title: Spectroscopy

text: The central theory of spectroscopy is that light is made of different wavelengths and that each wavelength corresponds to a different frequency. The importance of spectroscopy is centered around the fact that every different element in the periodic table has a unique light spectrum described by the frequencies of light it emits or absorbs consistently appearing in the same part of the electromagnetic spectrum when that light is diffracted. This opened up an entire field of study with anything that contains atoms which is all matter. Spectroscopy is the key to understanding the atomic properties of all matter. As such spectroscopy opened up many new sub-fields of science yet undiscovered. The idea that each atomic element has its unique spectral signature enabled spectroscopy to be used in a broad number of fields each with a specific goal achieved by different spectroscopic procedures. The National Institute of Standards and Technology maintains a public Atomic Spectra Database that is continually updated with precise measurements.


title: Spectroscopy

text: The broadening of the field of spectroscopy is due to the fact that any part of the electromagnetic spectrum may be used to analyze a sample from the infrared to the ultraviolet telling scientists different properties about the very same sample. For instance in chemical analysis, the most common types of spectroscopy include atomic spectroscopy, infrared spectroscopy, ultraviolet and visible spectroscopy, Raman spectroscopy and nuclear magnetic resonance. In nuclear magnetic resonance, the theory behind it is that frequency is analogous to resonance and its corresponding resonant frequency. Resonances by the frequency were first characterized in mechanical systems such as pendulums which have a frequency of motion noted famously by Galileo.


title: Spectroscopy

text: Spectroscopy is a sufficiently broad field that many sub-disciplines exist, each with numerous implementations of specific spectroscopic techniques. The various implementations and techniques can be classified in several ways.


title: Spectroscopy

text: The types of spectroscopy are distinguished by the type of radiative energy involved in the interaction. In many applications, the spectrum is determined by measuring changes in the intensity or frequency of this energy. The types of radiative energy studied include:


title: Spectroscopy

text: The types of spectroscopy also can be distinguished by the nature of the interaction between the energy and the material. These interactions include:


title: Spectroscopy

text: Spectroscopic studies are designed so that the radiant energy interacts with specific types of matter.


title: Spectroscopy

text: Atomic spectroscopy was the first application of spectroscopy developed. Atomic absorption spectroscopy and atomic emission spectroscopy involve visible and ultraviolet light. These absorptions and emissions, often referred to as atomic spectral lines, are due to electronic transitions of outer shell electrons as they rise and fall from one electron orbit to another. Atoms also have distinct x-ray spectra that are attributable to the excitation of inner shell electrons to excited states.


title: Spectroscopy

text: Atoms of different elements have distinct spectra and therefore atomic spectroscopy allows for the identification and quantitation of a sample's elemental composition. After inventing the spectroscope, Robert Bunsen and Gustav Kirchhoff discovered new elements by observing their emission spectra. Atomic absorption lines are observed in the solar spectrum and referred to as Fraunhofer lines after their discoverer. A comprehensive explanation of the hydrogen spectrum was an early success of quantum mechanics and explained the Lamb shift observed in the hydrogen spectrum, which further led to the development of quantum electrodynamics.


title: Spectroscopy

text: Modern implementations of atomic spectroscopy for studying visible and ultraviolet transitions include flame emission spectroscopy, inductively coupled plasma atomic emission spectroscopy, glow discharge spectroscopy, microwave induced plasma spectroscopy, and spark or arc emission spectroscopy. Techniques for studying x-ray spectra include X-ray spectroscopy and X-ray fluorescence.


title: Spectroscopy

text: The combination of atoms into molecules leads to the creation of unique types of energetic states and therefore unique spectra of the transitions between these states. Molecular spectra can be obtained due to electron spin states (electron paramagnetic resonance), molecular rotations, molecular vibration, and electronic states. Rotations are collective motions of the atomic nuclei and typically lead to spectra in the microwave and millimeter-wave spectral regions. Rotational spectroscopy and microwave spectroscopy are synonymous. Vibrations are relative motions of the atomic nuclei and are studied by both infrared and Raman spectroscopy. Electronic excitations are studied using visible and ultraviolet spectroscopy as well as fluorescence spectroscopy.


title: Spectroscopy

text: Studies in molecular spectroscopy led to the development of the first maser and contributed to the subsequent development of the laser.


title: Spectroscopy

text: The combination of atoms or molecules into crystals or other extended forms leads to the creation of additional energetic states. These states are numerous and therefore have a high density of states. This high density often makes the spectra weaker and less distinct, i.e., broader. For instance, blackbody radiation is due to the thermal motions of atoms and molecules within a material. Acoustic and mechanical responses are due to collective motions as well.


title: Spectroscopy

text: Pure crystals, though, can have distinct spectral transitions, and the crystal arrangement also has an effect on the observed molecular spectra. The regular lattice structure of crystals also scatters x-rays, electrons or neutrons allowing for crystallographic studies.


title: Spectroscopy

text: Nuclei also have distinct energy states that are widely separated and lead to gamma ray spectra. Distinct nuclear spin states can have their energy separated by a magnetic field, and this allows for nuclear magnetic resonance spectroscopy.


title: Spectroscopy

text: There are several applications of spectroscopy in the fields of medicine, physics, chemistry, and astronomy. Taking advantage of the properties of absorbance and with astronomy emission, spectroscopy can be used to identify certain states of nature. The uses of spectroscopy in so many different fields and for so many different applications has caused specialty scientific subfields. Such examples include:


title: Spectroscopy

text: The history of spectroscopy began with Isaac Newton's optics experiments (1666–1672). According to Andrew Fraknoi and David Morrison, "In 1672, in the first paper that he submitted to the Royal Society, Isaac Newton described an experiment in which he permitted sunlight to pass through a small hole and then through a prism. Newton found that sunlight, which looks white to us, is actually made up of a mixture of all the colors of the rainbow." Newton applied the word "spectrum" to describe the rainbow of colors that combine to form white light and that are revealed when the white light is passed through a prism.


title: Spectroscopy

text: Fraknoi and Morrison state that "In 1802, William Hyde Wollaston built an improved spectrometer that included a lens to focus the Sun's spectrum on a screen. Upon use, Wollaston realized that the colors were not spread uniformly, but instead had missing patches of colors, which appeared as dark bands in the spectrum." During the early 1800s, Joseph von Fraunhofer made experimental advances with dispersive spectrometers that enabled spectroscopy to become a more precise and quantitative scientific technique. Since then, spectroscopy has played and continues to play a significant role in chemistry, physics, and astronomy. Per Fraknoi and Morrison, "Later, in 1815, German physicist Joseph Fraunhofer also examined the solar spectrum, and found about 600 such dark lines (missing colors), are now known as Fraunhofer lines, or Absorption lines."


title: Spectroscopy

text: In quantum mechanical systems, the analogous resonance is a coupling of two quantum mechanical stationary states of one system, such as an atom, via an oscillatory source of energy such as a photon. The coupling of the two states is strongest when the energy of the source matches the energy difference between the two states. The energy of a photon is related to its frequency by where is Planck's constant, and so a spectrum of the system response vs. photon frequency will peak at the resonant frequency or energy. Particles such as electrons and neutrons have a comparable relationship, the de Broglie relations, between their kinetic energy and their wavelength and frequency and therefore can also excite resonant interactions.


title: Spectroscopy

text: Spectra of atoms and molecules often consist of a series of spectral lines, each one representing a resonance between two different quantum states. The explanation of these series, and the spectral patterns associated with them, were one of the experimental enigmas that drove the development and acceptance of quantum mechanics. The hydrogen spectral series in particular was first successfully explained by the Rutherford–Bohr quantum model of the hydrogen atom. In some cases spectral lines are well separated and distinguishable, but spectral lines can also overlap and appear to be a single transition if the density of energy states is high enough. Named series of lines include the principal, sharp, diffuse and fundamental series.


title: Peptide

text: Peptides (, ) are short chains of amino acids linked by peptide bonds. Long chains of amino acids are called proteins. Chains of fewer than twenty amino acids are called oligopeptides, and include dipeptides, tripeptides, and tetrapeptides.


title: Peptide

text: A polypeptide is a longer, continuous, unbranched peptide chain. Hence, peptides fall under the broad chemical classes of biological polymers and oligomers, alongside nucleic acids, oligosaccharides, polysaccharides, and others.


title: Peptide

text: A polypeptide that contains more than approximately 50 amino acids is known as a protein. Proteins consist of one or more polypeptides arranged in a biologically functional way, often bound to ligands such as coenzymes and cofactors, or to another protein or other macromolecule such as DNA or RNA, or to complex macromolecular assemblies.


title: Peptide

text: Amino acids that have been incorporated into peptides are termed residues. A water molecule is released during formation of each amide bond. All peptides except cyclic peptides have an N-terminal (amine group) and C-terminal (carboxyl group) residue at the end of the peptide (as shown for the tetrapeptide in the image).


title: Peptide

text: There are numerous types of peptides that have been classified according to their sources and functions. According to the Handbook of Biologically Active Peptides, some groups of peptides include plant peptides, bacterial/antibiotic peptides, fungal peptides, invertebrate peptides, amphibian/skin peptides, venom peptides, cancer/anticancer peptides, vaccine peptides, immune/inflammatory peptides, brain peptides, endocrine peptides, ingestive peptides, gastrointestinal peptides, cardiovascular peptides, renal peptides, respiratory peptides, opiate peptides, neurotrophic peptides, and blood–brain peptides.


title: Peptide

text: Some ribosomal peptides are subject to proteolysis. These function, typically in higher organisms, as hormones and signaling molecules. Some microbes produce peptides as antibiotics, such as microcins and bacteriocins.


title: Peptide

text: Peptides frequently have post-translational modifications such as phosphorylation, hydroxylation, sulfonation, palmitoylation, glycosylation, and disulfide formation. In general, peptides are linear, although lariat structures have been observed. More exotic manipulations do occur, such as racemization of L-amino acids to D-amino acids in platypus venom.


title: Peptide

text: "Nonribosomal peptides" are assembled by enzymes, not the ribosome. A common non-ribosomal peptide is glutathione, a component of the antioxidant defenses of most aerobic organisms. Other nonribosomal peptides are most common in unicellular organisms, plants, and fungi and are synthesized by modular enzyme complexes called "nonribosomal peptide synthetases".


title: Peptide

text: These complexes are often laid out in a similar fashion, and they can contain many different modules to perform a diverse set of chemical manipulations on the developing product. These peptides are often cyclic and can have highly complex cyclic structures, although linear nonribosomal peptides are also common. Since the system is closely related to the machinery for building fatty acids and polyketides, hybrid compounds are often found. The presence of oxazoles or thiazoles often indicates that the compound was synthesized in this fashion.


title: Peptide

text: "" are derived from animal milk or meat digested by proteolysis. In addition to containing small peptides, the resulting material includes fats, metals, salts, vitamins, and many other biological compounds. Peptones are used in nutrient media for growing bacteria and fungi.


title: Peptide

text: "Peptide fragments" refer to fragments of proteins that are used to identify or quantify the source protein. Often these are the products of enzymatic degradation performed in the laboratory on a controlled sample, but can also be forensic or paleontological samples that have been degraded by natural effects.


title: Peptide

text: The peptide families in this section are ribosomal peptides, usually with hormonal activity. All of these peptides are synthesized by cells as longer "propeptides" or "proproteins" and truncated prior to exiting the cell. They are released into the bloodstream where they perform their signaling functions.


title: Peptide

text: Several terms related to peptides have no strict length definitions, and there is often overlap in their usage:


title: Peptide

text: Peptides and proteins are often described by the number of amino acids in their chain, e.g. a protein with 158 amino acids may be described as a "158 amino-acid-long protein".


title: Peptide

text: The same words are also used to describe a group of residues in a larger polypeptide ("e.g.", RGD motif).


title: Tongue

text: The tongue is a muscular organ in the mouth of a typical tetrapod. It manipulates food for mastication and swallowing as part of the digestive process, and is the primary organ of taste. The tongue's upper surface (dorsum) is covered by taste buds housed in numerous lingual papillae. It is sensitive and kept moist by saliva and is richly supplied with nerves and blood vessels. The tongue also serves as a natural means of cleaning the teeth. A major function of the tongue is the enabling of speech in humans and vocalization in other animals.


title: Tongue

text: The human tongue is divided into two parts, an oral part at the front and a pharyngeal part at the back. The left and right sides are also separated along most of its length by a vertical section of fibrous tissue (the lingual septum) that results in a groove, the median sulcus, on the tongue's surface.


title: Tongue

text: There are two groups of muscles of the tongue. The four intrinsic muscles alter the shape of the tongue and are not attached to bone. The four paired extrinsic muscles change the position of the tongue and are anchored to bone.


title: Tongue

text: The word tongue derives from the Old English "tunge", which comes from Proto-Germanic *"tungōn". It has cognates in other Germanic languages—for example "tonge" in West Frisian, "tong" in Dutch and Afrikaans, "Zunge" in German, "tunge" in Danish and Norwegian, and "tunga" in Icelandic, Faroese and Swedish. The "ue" ending of the word seems to be a fourteenth-century attempt to show "proper pronunciation", but it is "neither etymological nor phonetic". Some used the spelling "tunge" and "tonge" as late as the sixteenth century.


title: Tongue

text: The tongue is a muscular hydrostat that forms part of the floor of the oral cavity. The left and right sides of the tongue are separated by a vertical section of fibrous tissue known as the lingual septum. This division is along the length of the tongue save for the very back of the pharyngeal part and is visible as a groove called the median sulcus. The human tongue is divided into anterior and posterior parts by the terminal sulcus which is a V-shaped groove. The apex of the terminal sulcus is marked by a blind foramen, the foramen cecum, which is a remnant of the median thyroid diverticulum in early embryonic development. The anterior "oral" part is the visible part situated at the front and makes up roughly two-thirds the length of the tongue. The posterior "pharyngeal" part is the part closest to the throat, roughly one-third of its length. These parts differ in terms of their embryological development and nerve supply.


title: Tongue

text: The anterior tongue is, at its apex, thin and narrow. It is directed forward against the lingual surfaces of the lower incisor teeth. The posterior part is, at its root, directed backward, and connected with the hyoid bone by the hyoglossi and genioglossi muscles and the hyoglossal membrane, with the epiglottis by three glossoepiglottic folds of mucous membrane, with the soft palate by the glossopalatine arches, and with the pharynx by the superior pharyngeal constrictor muscle and the mucous membrane. It also forms the anterior wall of the oropharynx.


title: Tongue

text: The average length of the human tongue from the oropharynx to the tip is 10 cm. The average weight of the human tongue from adult males is 70g and for adult females 60g.


title: Tongue

text: In phonetics and phonology, a distinction is made between the tip of the tongue and the blade (the portion just behind the tip). Sounds made with the tongue tip are said to be apical, while those made with the tongue blade are said to be laminal.


title: Tongue

text: The upper surface of the tongue is called the dorsum, and is divided by a groove into symmetrical halves by the median sulcus. The foramen cecum marks the end of this division (at about 2.5 cm from the root of the tongue) and the beginning of the terminal sulcus. The foramen cecum is also the point of attachment of the thyroglossal duct and is formed during the descent of the thyroid diverticulum in embryonic development.


title: Tongue

text: The terminal sulcus is a shallow groove that runs forward as a shallow groove in a "V" shape from the foramen cecum, forwards and outwards to the margins (borders) of the tongue. The terminal sulcus divides the tongue into a posterior pharyngeal part and an anterior oral part. The pharyngeal part is supplied by the glossopharyngeal nerve and the oral part is supplied by the lingual nerve (a branch of the mandibular branch (V3) of the trigeminal nerve) for somatosensory perception and by the chorda tympani (a branch of the facial nerve) for taste perception.


title: Tongue

text: On the undersurface of the tongue is a fold of mucous membrane called the frenulum that tethers the tongue at the midline to the floor of the mouth. On either side of the frenulum are small prominences called sublingual caruncles that the major salivary submandibular glands drain into.


title: Tongue

text: The eight muscles of the human tongue are classified as either "intrinsic" or "extrinsic". The four intrinsic muscles act to change the shape of the tongue, and are not attached to any bone. The four extrinsic muscles act to change the position of the tongue, and are anchored to bone.


title: Tongue

text: The four extrinsic muscles originate from bone and extend to the tongue. They are the genioglossus, the hyoglossus (often including the chondroglossus) the styloglossus, and the palatoglossus. Their main functions are altering the tongue's position allowing for protrusion, retraction, and side-to-side movement.


title: Tongue

text: The genioglossus arises from the mandible and protrudes the tongue. It is also known as the tongue's "safety muscle" since it is the only muscle that propels the tongue forward.


title: Tongue

text: The hyoglossus, arises from the hyoid bone and retracts and depresses the tongue. The chondroglossus is often included with this muscle.


title: Tongue

text: The styloglossus arises from the styloid process of the temporal bone and draws the sides of the tongue up to create a trough for swallowing.


title: Tongue

text: The palatoglossus arises from the palatine aponeurosis, and depresses the soft palate, moves the "palatoglossal fold" towards the midline, and elevates the back of the tongue during swallowing.


title: Tongue

text: Four paired intrinsic muscles of the tongue originate and insert within the tongue, running along its length. They are the superior longitudinal muscle, the inferior longitudinal muscle, the vertical muscle, and the transverse muscle. These muscles alter the shape of the tongue by lengthening and shortening it, curling and uncurling its apex and edges as in tongue rolling, and flattening and rounding its surface. This provides shape and helps facilitate speech, swallowing, and eating.


title: Tongue

text: The superior longitudinal muscle runs along the upper surface of the tongue under the mucous membrane, and elevates, assists in retraction of, or deviates the tip of the tongue. It originates near the epiglottis, at the hyoid bone, from the median fibrous septum.


title: Tongue

text: The inferior longitudinal muscle lines the sides of the tongue, and is joined to the styloglossus muscle.


title: Tongue

text: The vertical muscle is located in the middle of the tongue, and joins the superior and inferior longitudinal muscles.


title: Tongue

text: The transverse muscle divides the tongue at the middle, and is attached to the mucous membranes that run along the sides.


title: Tongue

text: The tongue receives its blood supply primarily from the lingual artery, a branch of the external carotid artery. The lingual veins drain into the internal jugular vein. The floor of the mouth also receives its blood supply from the lingual artery. There is also a secondary blood supply to the root of tongue from the tonsillar branch of the facial artery and the ascending pharyngeal artery.


title: Tongue

text: An area in the neck sometimes called the Pirogov triangle is formed by the intermediate tendon of the digastric muscle, the posterior border of the mylohyoid muscle, and the hypoglossal nerve. The lingual artery is a good place to stop severe hemorrhage from the tongue.


title: Tongue

text: Innervation of the tongue consists of motor fibers, special sensory fibers for taste, and general sensory fibers for sensation.


title: Tongue

text: Innervation of taste and sensation is different for the anterior and posterior part of the tongue because they are derived from different embryological structures (pharyngeal arch 1 and pharyngeal arches 3 and 4, respectively).


title: Tongue

text: The tip of tongue drains to the submental nodes. The left and right halves of the anterior two-thirds of the tongue drains to submandibular lymph nodes, while the posterior one-third of the tongue drains to the jugulo-omohyoid nodes.


title: Tongue

text: The upper surface of the tongue is covered in masticatory mucosa, a type of oral mucosa which is of keratinized stratified squamous epithelium. Embedded in this are numerous papillae, some of which house the taste buds and their taste receptors. The lingual papillae consist of filiform, fungiform, vallate and foliate papillae, and only the filiform papillae are not associated with any taste buds.


title: Tongue

text: The tongue can divide itself in dorsal and ventral surface. The dorsal surface is a stratified squamous keratinized epithelium which is characterized by numerous mucosal projections called papillae. The lingual papillae covers the dorsal side of the tongue towards the front of the terminal groove. The ventral surface is stratified squamous non-keratinized epithelium which is smooth.


title: Tongue

text: The tongue begins to develop in the fourth week of embryonic development from a median swelling – the median tongue bud (tuberculum impar) of the first pharyngeal arch.


title: Tongue

text: In the fifth week a pair of lateral lingual swellings, one on the right side and one on the left, form on the first pharyngeal arch. These lingual swellings quickly expand and cover the median tongue bud. They form the anterior part of the tongue that makes up two-thirds of the length of the tongue, and continue to develop through prenatal development. The line of their fusion is marked by the median sulcus.


title: Tongue

text: In the fourth week a swelling appears from the second pharyngeal arch, in the midline, called the copula. During the fifth and sixth weeks the copula is overgrown by a swelling from the third and fourth arches (mainly from the third arch) called the hypopharyngeal eminence, and this develops into the posterior part of the tongue (the other third and the posterior most part of the tongue is developed from the fourth pharyngeal arch ). The hypopharyngeal eminence develops mainly by the growth of endoderm from the third pharyngeal arch. The boundary between the two parts of the tongue, the anterior from the first arch and the posterior from the third arch is marked by the terminal sulcus. The terminal sulcus is shaped like a "V" with the tip of the V situated posteriorly. At the tip of the terminal sulcus is the foramen cecum, which is the point of attachment of the thyroglossal duct where the embryonic thyroid begins to descend.


title: Tongue

text: Chemicals that stimulate taste receptor cells are known as tastants. Once a tastant is dissolved in saliva, it can make contact with the plasma membrane of the gustatory hairs, which are the sites of taste transduction.


title: Tongue

text: The tongue is equipped with many taste buds on its dorsal surface, and each taste bud is equipped with taste receptor cells that can sense particular classes of tastes. Distinct types of taste receptor cells respectively detect substances that are sweet, bitter, salty, sour, spicy, or taste of umami. Umami receptor cells are the least understood and accordingly are the type most intensively under research. Tongue map is a myth that is widely taught in schools.


title: Tongue

text: The tongue is an important accessory organ in the digestive system. The tongue is used for crushing food against the hard palate, during mastication and manipulation of food for softening prior to swallowing. The epithelium on the tongue's upper, or dorsal surface is keratinised. Consequently, the tongue can grind against the hard palate without being itself damaged or irritated.


title: Tongue

text: The tongue is one of the primary articulators in the production of speech, and this is facilitated by both the extrinsic muscles that move the tongue and the intrinsic muscles that change its shape. Specifically, different vowels are articulated by changing the tongue's height and retraction to alter the resonant properties of the vocal tract. These resonant properties amplify specific harmonic frequencies (formants) that are different for each vowel, while attenuating other harmonics. For example, [a] is produced with the tongue lowered and centered and [i] is produced with the tongue raised and fronted. Consonants are articulated by constricting airflow through the vocal tract, and many consonants feature a constriction between the tongue and some other part of the vocal tract. For example alveolar consonants like [s] and [n] are articulated with the tongue against the alveolar ridge, while velar consonants like [k] and [g] are articulated with the tongue dorsum against the soft palate (velum). Tongue shape is also relevant to speech articulation, for example in retroflex consonants, where the tip of the tongue is curved backward.


title: Tongue

text: The tongue plays a role in physical intimacy and sexuality. The tongue is part of the erogenous zone of the mouth and can be used in intimate contact, as in the French kiss and in oral sex. The tongue can be used for stimulating the clitoris and other areas of the vulva.


title: Tongue

text: A congenital disorder of the tongue is that of ankyloglossia also known as "tongue-tie". The tongue is "tied" to the floor of the mouth by a very short and thickened frenulum and this affects speech, eating, and swallowing.


title: Tongue

text: The tongue is prone to several pathologies including glossitis and other inflammations such as geographic tongue, and median rhomboid glossitis; burning mouth syndrome, oral hairy leukoplakia, oral candidiasis (thrush), black hairy tongue , bifid tongue (due to failure in fusion of two lingual swellings of first pharyngeal arch) and fissured tongue.


title: Tongue

text: There are several types of oral cancer that mainly affect the tongue. Mostly these are squamous cell carcinomas.


title: Tongue

text: Food debris, desquamated epithelial cells and bacteria often form a visible tongue coating. This coating has been identified as a major factor contributing to bad breath (halitosis), which can be managed by using a tongue cleaner.


title: Tongue

text: The sublingual region underneath the front of the tongue is an ideal location for the administration of certain medications into the body. The oral mucosa is very thin underneath the tongue, and is underlain by a plexus of veins. The sublingual route takes advantage of the highly vascular quality of the oral cavity, and allows for the speedy application of medication into the cardiovascular system, bypassing the gastrointestinal tract. This is the only convenient and efficacious route of administration (apart from Intravenous therapy) of nitroglycerin to a patient suffering chest pain from angina pectoris.


title: Tongue

text: The muscles of the tongue evolved in amphibians from occipital somites. Most amphibians show a proper tongue after their metamorphosis. As a consequence most vertebrate animals—amphibians, reptiles, birds, and mammals—have tongues (the frog family of pipids lack tongue). In mammals such as dogs and cats, the tongue is often used to clean the fur and body by licking. The tongues of these species have a very rough texture which allows them to remove oils and parasites. Some dogs have a tendency to consistently lick a part of their foreleg which can result in a skin condition known as a lick granuloma. A dog's tongue also acts as a heat regulator. As a dog increases its exercise the tongue will increase in size due to greater blood flow. The tongue hangs out of the dog's mouth and the moisture on the tongue will work to cool the bloodflow.


title: Tongue

text: Some animals have tongues that are specially adapted for catching prey. For example, chameleons, frogs, pangolins and anteaters have prehensile tongues.


title: Tongue

text: Other animals may have organs that are analogous to tongues, such as a butterfly's proboscis or a radula on a mollusc, but these are not homologous with the tongues found in vertebrates and often have little resemblance in function. For example, butterflies do not lick with their proboscides; they suck through them, and the proboscis is not a single organ, but two jaws held together to form a tube. Many species of fish have small folds at the base of their mouths that might informally be called tongues, but they lack a muscular structure like the true tongues found in most tetrapods.


title: Tongue

text: The tongue can be used as a metonym for "language". For example, the New Testament of the Bible, in the Book of Acts of the Apostles, Jesus' disciples on the Day of Pentecost received a type of spiritual gift: "there appeared unto them cloven tongues like as of fire, and it sat upon each of them. And they were all filled with the Holy Ghost, and began to speak with other tongues ...", which amazed the crowd of Jewish people in Jerusalem, who were from various parts of the Roman Empire but could now understand what was being preached. The phrase "mother tongue" is used as a child's first language. Many languages have the same word for "tongue" and "language", as did the English language before the Middle Ages.


title: Tongue

text: A common temporary failure in word retrieval from memory is referred to as the "tip-of-the-tongue" phenomenon. The expression "tongue in cheek" refers to a statement that is not to be taken entirely seriously – something said or done with subtle ironic or sarcastic humour. A "tongue twister" is a phrase made specifically to be very difficult to pronounce. Aside from being a medical condition, "tongue-tied" means being unable to say what you want due to confusion or restriction. The phrase "cat got your tongue" refers to when a person is speechless. To "bite one's tongue" is a phrase which describes holding back an opinion to avoid causing offence. A "slip of the tongue" refers to an unintentional utterance, such as a Freudian slip. The "gift of tongues" refers to when one is uncommonly gifted to be able to speak in a foreign language, often as a type of spiritual gift. Speaking in tongues is a common phrase used to describe "glossolalia", which is to make smooth, language-resembling sounds that is no true spoken language itself. A deceptive person is said to have a forked tongue, and a smooth-talking person said to have a .


title: Tongue

text: Sticking one's tongue out at someone is considered a childish gesture of rudeness or defiance in many countries; the act may also have sexual connotations, depending on the way in which it is done. However, in Tibet it is considered a greeting. In 2009, a farmer from Fabriano, Italy, was convicted and fined by the country's highest court for sticking his tongue out at a neighbor with whom he had been arguing. Proof of the affront had been captured with a cell phone camera.


title: Tongue

text: Tongue piercing and splitting have become more common in western countries in recent decades. In one study, one-fifth of young adults were found to have at least one type of oral piercing, most commonly the tongue.


title: Tongue

text: The tongues of some animals are consumed and sometimes considered delicacies. Hot tongue sandwiches are frequently found on menus in kosher delicatessens in America. Taco de lengua ("lengua" being Spanish for tongue) is a taco filled with beef tongue, and is especially popular in Mexican cuisine. As part of Colombian gastronomy, Tongue in Sauce (Lengua en Salsa), is a dish prepared by frying the tongue, adding tomato sauce, onions and salt. Tongue can also be prepared as birria. Pig and beef tongue are consumed in Chinese cuisine. Duck tongues are sometimes employed in Szechuan dishes, while lamb's tongue is occasionally employed in Continental and contemporary American cooking. Fried cod "tongue" is a relatively common part of fish meals in Norway and Newfoundland. In Argentina and Uruguay cow tongue is cooked and served in vinegar ("lengua a la vinagreta"). In the Czech Republic and Poland, a pork tongue is considered a delicacy, and there are many ways of preparing it. In Eastern Slavic countries, pork and beef tongues are commonly consumed, boiled and garnished with horseradish or jelled; beef tongues fetch a significantly higher price and are considered more of a delicacy. In Alaska, cow tongues are among the more common. Both cow and moose tongues are popular toppings on open-top-sandwiches in Norway, the latter usually amongst hunters.


title: Tongue

text: Tongues of seals and whales have been eaten, sometimes in large quantities, by sealers and whalers, and in various times and places have been sold for food on shore.


title: Cell cycle

text: The cell cycle, or cell-division cycle, is the series of events that take place in a cell that cause it to divide into two daughter cells. These events include the duplication of its DNA (DNA replication) and some of its organelles, and subsequently the partitioning of its cytoplasm, chromosomes and other components into two daughter cells in a process called cell division.


title: Cell cycle

text: In cells with nuclei (eukaryotes, i.e., animal, plant, fungal, and protist cells), the cell cycle is divided into two main stages: interphase and the mitotic (M) phase (including mitosis and cytokinesis). During interphase, the cell grows, accumulating nutrients needed for mitosis, and replicates its DNA and some of its organelles. During the mitotic phase, the replicated chromosomes, organelles, and cytoplasm separate into two new daughter cells. To ensure the proper replication of cellular components and division, there are control mechanisms known as cell cycle checkpoints after each of the key steps of the cycle that determine if the cell can progress to the next phase.


title: Cell cycle

text: In cells without nuclei (prokaryotes, i.e., bacteria and archaea), the cell cycle is divided into the B, C, and D periods. The B period extends from the end of cell division to the beginning of DNA replication. DNA replication occurs during the C period. The D period refers to the stage between the end of DNA replication and the splitting of the bacterial cell into two daughter cells.


title: Cell cycle

text: In single-celled organisms, a single cell-division cycle is how the organism replicates itself. In multicellular organisms such as plants and animals, a series of cell-division cycles is how the organism develops from a single-celled fertilized egg into a mature organism, and is also the process by which hair, skin, blood cells, and some internal organs are regenerated and healed (with possible exception of nerves; see nerve damage). After cell division, each of the daughter cells begin the interphase of a new cell cycle. Although the various stages of interphase are not usually morphologically distinguishable, each phase of the cell cycle has a distinct set of specialized biochemical processes that prepare the cell for initiation of the cell division.


title: Cell cycle

text: The eukaryotic cell cycle consists of four distinct phases: G phase, S phase (synthesis), G phase (collectively known as interphase) and M phase (mitosis and cytokinesis). M phase is itself composed of two tightly coupled processes: mitosis, in which the cell's nucleus divides, and cytokinesis, in which the cell's cytoplasm divides forming two daughter cells. Activation of each phase is dependent on the proper progression and completion of the previous one. Cells that have temporarily or reversibly stopped dividing are said to have entered a state of quiescence called G phase.


title: Cell cycle

text: After cell division, each of the daughter cells begin the interphase of a new cycle. Although the various stages of interphase are not usually morphologically distinguishable, each phase of the cell cycle has a distinct set of specialized biochemical processes that prepare the cell for initiation of cell division.


title: Cell cycle

text: G is a resting phase where the cell has left the cycle and has stopped dividing. The cell cycle starts with this phase. Non-proliferative (non-dividing) cells in multicellular eukaryotes generally enter the quiescent G state from G and may remain quiescent for long periods of time, possibly indefinitely (as is often the case for neurons). This is very common for cells that are fully differentiated. Some cells enter the G phase semi-permanently and are considered post-mitotic, e.g., some liver, kidney, and stomach cells. Many cells do not enter G and continue to divide throughout an organism's life, e.g., epithelial cells.


title: Cell cycle

text: The word "post-mitotic" is sometimes used to refer to both quiescent and senescent cells. Cellular senescence occurs in response to DNA damage and external stress and usually constitutes an arrest in G. Cellular senescence may make a cell's progeny nonviable; it is often a biochemical alternative to the self-destruction of such a damaged cell by apoptosis.


title: Cell cycle

text: Interphase represent the phase between two successive M phases. Interphase is a series of changes that takes place in a newly formed cell and its nucleus before it becomes capable of division again. It is also called preparatory phase or intermitosis. Typically interphase lasts for at least 91% of the total time required for the cell cycle.


title: Cell cycle

text: Interphase proceeds in three stages, G, S, and G, followed by the cycle of mitosis and cytokinesis. The cell's nuclear DNA contents are duplicated during S phase.


title: Cell cycle

text: The first phase within interphase, from the end of the previous M phase until the beginning of DNA synthesis, is called G (G indicating "gap"). It is also called the growth phase. During this phase, the biosynthetic activities of the cell, which are considerably slowed down during M phase, resume at a high rate. The duration of G is highly variable, even among different cells of the same species. In this phase, the cell increases its supply of proteins, increases the number of organelles (such as mitochondria, ribosomes), and grows in size. In G phase, a cell has three options.


title: Cell cycle

text: The deciding point is called check point (Restriction point). This check point is called the restriction point or START and is regulated by G/S cyclins, which cause transition from G to S phase. Passage through the G check point commits the cell to division.


title: Cell cycle

text: The ensuing S phase starts when DNA synthesis commences; when it is complete, all of the chromosomes have been replicated, i.e., each chromosome consists of two sister chromatids. Thus, during this phase, the amount of DNA in the cell has doubled, though the ploidy and number of chromosomes are unchanged. Rates of RNA transcription and protein synthesis are very low during this phase. An exception to this is histone production, most of which occurs during the S phase.


title: Cell cycle

text: G phase occurs after DNA replication and is a period of protein synthesis and rapid cell growth to prepare the cell for mitosis.


title: Cell cycle

text: During this phase microtubules begin to reorganize to form a spindle (preprophase). Before proceeding to mitotic phase, cells must be checked at the G checkpoint for any DNA damage within the chromosomes. The G checkpoint is mainly regulated by the tumor protein p53. If the DNA is damaged, p53 will either repair the DNA or trigger the apoptosis of the cell. If p53 is dysfunctional or mutated, cells with damaged DNA may continue through the cell cycle, leading to the development of cancer.


title: Cell cycle

text: The relatively brief "M phase" consists of nuclear division (karyokinesis). It is a relatively short period of the cell cycle. M phase is complex and highly regulated. The sequence of events is divided into phases, corresponding to the completion of one set of activities and the start of the next. These phases are sequentially known as:


title: Cell cycle

text: Mitosis is the process by which a eukaryotic cell separates the chromosomes in its cell nucleus into two identical sets in two nuclei. During the process of mitosis the pairs of chromosomes condense and attach to microtubules that pull the sister chromatids to opposite sides of the cell.


title: Cell cycle

text: Mitosis occurs exclusively in eukaryotic cells, but occurs in different ways in different species. For example, animal cells undergo an "open" mitosis, where the nuclear envelope breaks down before the chromosomes separate, while fungi such as "Aspergillus nidulans" and "Saccharomyces cerevisiae" (yeast) undergo a "closed" mitosis, where chromosomes divide within an intact cell nucleus.


title: Cell cycle

text: Mitosis is immediately followed by cytokinesis, which divides the nuclei, cytoplasm, organelles and cell membrane into two cells containing roughly equal shares of these cellular components. Mitosis and cytokinesis together define the division of the mother cell into two daughter cells, genetically identical to each other and to their parent cell. This accounts for approximately 10% of the cell cycle.


title: Cell cycle

text: Because cytokinesis usually occurs in conjunction with mitosis, "mitosis" is often used interchangeably with "M phase". However, there are many cells where mitosis and cytokinesis occur separately, forming single cells with multiple nuclei in a process called endoreplication. This occurs most notably among the fungi and slime molds, but is found in various groups. Even in animals, cytokinesis and mitosis may occur independently, for instance during certain stages of fruit fly embryonic development. Errors in mitosis can result in cell death through apoptosis or cause mutations that may lead to cancer.


title: Cell cycle

text: Regulation of the cell cycle involves processes crucial to the survival of a cell, including the detection and repair of genetic damage as well as the prevention of uncontrolled cell division. The molecular events that control the cell cycle are ordered and directional; that is, each process occurs in a sequential fashion and it is impossible to "reverse" the cycle.


title: Cell cycle

text: Two key classes of regulatory molecules, cyclins and cyclin-dependent kinases (CDKs), determine a cell's progress through the cell cycle. Leland H. Hartwell, R. Timothy Hunt, and Paul M. Nurse won the 2001 Nobel Prize in Physiology or Medicine for their discovery of these central molecules. Many of the genes encoding cyclins and CDKs are conserved among all eukaryotes, but in general, more complex organisms have more elaborate cell cycle control systems that incorporate more individual components. Many of the relevant genes were first identified by studying yeast, especially "Saccharomyces cerevisiae"; genetic nomenclature in yeast dubs many of these genes "cdc" (for "cell division cycle") followed by an identifying number, e.g. "cdc25" or "cdc20".


title: Cell cycle

text: Cyclins form the regulatory subunits and CDKs the catalytic subunits of an activated heterodimer; cyclins have no catalytic activity and CDKs are inactive in the absence of a partner cyclin. When activated by a bound cyclin, CDKs perform a common biochemical reaction called phosphorylation that activates or inactivates target proteins to orchestrate coordinated entry into the next phase of the cell cycle. Different cyclin-CDK combinations determine the downstream proteins targeted. CDKs are constitutively expressed in cells whereas cyclins are synthesised at specific stages of the cell cycle, in response to various molecular signals.


title: Cell cycle

text: Upon receiving a pro-mitotic extracellular signal, G cyclin-CDK complexes become active to prepare the cell for S phase, promoting the expression of transcription factors that in turn promote the expression of S cyclins and of enzymes required for DNA replication. The G cyclin-CDK complexes also promote the degradation of molecules that function as S phase inhibitors by targeting them for ubiquitination. Once a protein has been ubiquitinated, it is targeted for proteolytic degradation by the proteasome. However, results from a recent study of E2F transcriptional dynamics at the single-cell level argue that the role of G1 cyclin-CDK activities, in particular cyclin D-CDK4/6, is to tune the timing rather than the commitment of cell cycle entry.


title: Cell cycle

text: Active S cyclin-CDK complexes phosphorylate proteins that make up the pre-replication complexes assembled during G phase on DNA replication origins. The phosphorylation serves two purposes: to activate each already-assembled pre-replication complex, and to prevent new complexes from forming. This ensures that every portion of the cell's genome will be replicated once and only once. The reason for prevention of gaps in replication is fairly clear, because daughter cells that are missing all or part of crucial genes will die. However, for reasons related to gene copy number effects, possession of extra copies of certain genes is also deleterious to the daughter cells.


title: Cell cycle

text: Mitotic cyclin-CDK complexes, which are synthesized but inactivated during S and G phases, promote the initiation of mitosis by stimulating downstream proteins involved in chromosome condensation and mitotic spindle assembly. A critical complex activated during this process is a ubiquitin ligase known as the anaphase-promoting complex (APC), which promotes degradation of structural proteins associated with the chromosomal kinetochore. APC also targets the mitotic cyclins for degradation, ensuring that telophase and cytokinesis can proceed.


title: Cell cycle

text: Cyclin D is the first cyclin produced in the cells that enter the cell cycle, in response to extracellular signals (e.g. growth factors). Cyclin D levels stay low in resting cells that are not proliferating. Additionally, CDK4/6 and CDK2 are also inactive because CDK4/6 are bound by INK4 family members (e.g., p16), limiting kinase activity. Meanwhile, CDK2 complexes are inhibited by the CIP/KIP proteins such as p21 and p27, When it is time for a cell to enter the cell cycle, which is triggered by a mitogenic stimuli, levels of cyclin D increase. In response to this trigger, cyclin D binds to existing CDK4/6, forming the active cyclin D-CDK4/6 complex. Cyclin D-CDK4/6 complexes in turn mono-phosphorylates the retinoblastoma susceptibility protein (Rb) to pRb. The un-phosphorylated Rb tumour suppressor functions in inducing cell cycle exit and maintaining G0 arrest (senescence).


title: Cell cycle

text: In the last few decades, a model has been widely accepted whereby pRB proteins are inactivated by cyclin D-Cdk4/6-mediated phosphorylation. Rb has 14+ potential phosphorylation sites. Cyclin D-Cdk 4/6 progressively phosphorylates Rb to hyperphosphorylated state, which triggers dissociation of pRB–E2F complexes, thereby inducing G1/S cell cycle gene expression and progression into S phase.


title: Cell cycle

text: However, scientific observations from a recent study show that Rb is present in three types of isoforms: (1) un-phosphorylated Rb in G0 state; (2) mono-phosphorylated Rb, also referred to as "hypo-phosphorylated' or 'partially' phosphorylated Rb in early G1 state; and (3) inactive hyper-phosphorylated Rb in late G1 state. In early G1 cells, mono-phosphorylated Rb exits as 14 different isoforms, one of each has distinct E2F binding affinity. Rb has been found to associate with hundreds of different proteins and the idea that different mono-phosphorylated Rb isoforms have different protein partners was very appealing. A recent report confirmed that mono-phosphorylation controls Rb's association with other proteins and generates functional distinct forms of Rb. All different mono-phosphorylated Rb isoforms inhibit E2F transcriptional program and are able to arrest cells in G1-phase. Importantly, different mono-phosphorylated forms of RB have distinct transcriptional outputs that are extended beyond E2F regulation.


title: Cell cycle

text: In general, the binding of pRb to E2F inhibits the E2F target gene expression of certain G1/S and S transition genes including E-type cyclins. The partial phosphorylation of RB de-represses the Rb-mediated suppression of E2F target gene expression, begins the expression of cyclin E. The molecular mechanism that causes the cell switched to cyclin E activation is currently not known, but as cyclin E levels rise, the active cyclin E-CDK2 complex is formed, bringing Rb to be inactivated by hyper-phosphorylation. Hyperphosphorylated Rb is completely dissociated from E2F, enabling further expression of a wide range of E2F target genes are required for driving cells to proceed into S phase [1]. Recently, it has been identified that cyclin D-Cdk4/6 binds to a C-terminal alpha-helix region of Rb that is only distinguishable to cyclin D rather than other cyclins, cyclin E, A and B. This observation based on the structural analysis of Rb phosphorylation supports that Rb is phosphorylated in a different level through multiple Cyclin-Cdk complexes. This also makes feasible the current model of a simultaneous switch-like inactivation of all mono-phosphorylated Rb isoforms through one type of Rb hyper-phosphorylation mechanism. In addition, mutational analysis of the cyclin D- Cdk 4/6 specific Rb C-terminal helix shows that disruptions of cyclin D-Cdk 4/6 binding to Rb prevents Rb phosphorylation, arrests cells in G1, and bolsters Rb's functions in tumor suppressor. This cyclin-Cdk driven cell cycle transitional mechanism governs a cell committed to the cell cycle that allows cell proliferation. A cancerous cell growth often accompanies with deregulation of Cyclin D-Cdk 4/6 activity.


title: Cell cycle

text: The hyperphosphorylated Rb dissociates from the E2F/DP1/Rb complex (which was bound to the E2F responsive genes, effectively "blocking" them from transcription), activating E2F. Activation of E2F results in transcription of various genes like cyclin E, cyclin A, DNA polymerase, thymidine kinase, etc. Cyclin E thus produced binds to CDK2, forming the cyclin E-CDK2 complex, which pushes the cell from G to S phase (G/S, which initiates the G/M transition). Cyclin B-cdk1 complex activation causes breakdown of nuclear envelope and initiation of prophase, and subsequently, its deactivation causes the cell to exit mitosis. A quantitative study of E2F transcriptional dynamics at the single-cell level by using engineered fluorescent reporter cells provided a quantitative framework for understanding the control logic of cell cycle entry, challenging the canonical textbook model. Genes that regulate the amplitude of E2F accumulation, such as Myc, determine the commitment in cell cycle and S phase entry. G1 cyclin-CDK activities are not the driver of cell cycle entry. Instead, they primarily tune the timing of E2F increase, thereby modulating the pace of cell cycle progression.


title: Cell cycle

text: Two families of genes, the "cip/kip" ("CDK interacting protein/Kinase inhibitory protein") family and the INK4a/ARF ("In"hibitor of "K"inase 4/"A"lternative "R"eading "F"rame) family, prevent the progression of the cell cycle. Because these genes are instrumental in prevention of tumor formation, they are known as tumor suppressors.


title: Cell cycle

text: The "cip/kip" family includes the genes p21, p27 and p57. They halt the cell cycle in G phase by binding to and inactivating cyclin-CDK complexes. p21 is activated by p53 (which, in turn, is triggered by DNA damage e.g. due to radiation). p27 is activated by Transforming Growth Factor β (TGF β), a growth inhibitor.


title: Cell cycle

text: The INK4a/ARF family includes p16, which binds to CDK4 and arrests the cell cycle in G phase, and p14 which prevents p53 degradation.


title: Cell cycle

text: Synthetic inhibitors of Cdc25 could also be useful for the arrest of cell cycle and therefore be useful as antineoplastic and anticancer agents.


title: Cell cycle

text: Many human cancers possess the hyper-activated Cdk 4/6 activities. Given the observations of cyclin D-Cdk 4/6 functions, inhibition of Cdk 4/6 should result in preventing a malignant tumor from proliferating. Consequently, scientists have tried to invent the synthetic Cdk4/6 inhibitor as Cdk4/6 has been characterized to be a therapeutic target for anti-tumor effectiveness. Three Cdk4/6 inhibitors - palbociclib, ribociclib, and abemaciclib - currently received FDA approval for clinical use to treat advanced-stage or metastatic, hormone-receptor-positive (HR-positive, HR+), HER2-negative (HER2-) breast cancer. For example, palbociclib is an orally active CDK4/6 inhibitor which has demonstrated improved outcomes for ER-positive/HER2-negative advanced breast cancer. The main side effect is neutropenia which can be managed by dose reduction.


title: Cell cycle

text: Cdk4/6 targeted therapy will only treat cancer types where Rb is expressed. Cancer cells with loss of Rb have primary resistance to Cdk4/6 inhibitors.


title: Cell cycle

text: Current evidence suggests that a semi-autonomous transcriptional network acts in concert with the CDK-cyclin machinery to regulate the cell cycle. Several gene expression studies in "Saccharomyces cerevisiae" have identified 800–1200 genes that change expression over the course of the cell cycle. They are transcribed at high levels at specific points in the cell cycle, and remain at lower levels throughout the rest of the cycle. While the set of identified genes differs between studies due to the computational methods and criteria used to identify them, each study indicates that a large portion of yeast genes are temporally regulated.


title: Cell cycle

text: Many periodically expressed genes are driven by transcription factors that are also periodically expressed. One screen of single-gene knockouts identified 48 transcription factors (about 20% of all non-essential transcription factors) that show cell cycle progression defects. Genome-wide studies using high throughput technologies have identified the transcription factors that bind to the promoters of yeast genes, and correlating these findings with temporal expression patterns have allowed the identification of transcription factors that drive phase-specific gene expression. The expression profiles of these transcription factors are driven by the transcription factors that peak in the prior phase, and computational models have shown that a CDK-autonomous network of these transcription factors is sufficient to produce steady-state oscillations in gene expression).


title: Cell cycle

text: Experimental evidence also suggests that gene expression can oscillate with the period seen in dividing wild-type cells independently of the CDK machinery. Orlando "et al." used microarrays to measure the expression of a set of 1,271 genes that they identified as periodic in both wild type cells and cells lacking all S-phase and mitotic cyclins ("clb1,2,3,4,5,6"). Of the 1,271 genes assayed, 882 continued to be expressed in the cyclin-deficient cells at the same time as in the wild type cells, despite the fact that the cyclin-deficient cells arrest at the border between G and S phase. However, 833 of the genes assayed changed behavior between the wild type and mutant cells, indicating that these genes are likely directly or indirectly regulated by the CDK-cyclin machinery. Some genes that continued to be expressed on time in the mutant cells were also expressed at different levels in the mutant and wild type cells. These findings suggest that while the transcriptional network may oscillate independently of the CDK-cyclin oscillator, they are coupled in a manner that requires both to ensure the proper timing of cell cycle events. Other work indicates that phosphorylation, a post-translational modification, of cell cycle transcription factors by Cdk1 may alter the localization or activity of the transcription factors in order to tightly control timing of target genes.


title: Cell cycle

text: While oscillatory transcription plays a key role in the progression of the yeast cell cycle, the CDK-cyclin machinery operates independently in the early embryonic cell cycle. Before the midblastula transition, zygotic transcription does not occur and all needed proteins, such as the B-type cyclins, are translated from maternally loaded mRNA.


title: Cell cycle

text: Analyses of synchronized cultures of "Saccharomyces cerevisiae" under conditions that prevent DNA replication initiation without delaying cell cycle progression showed that origin licensing decreases the expression of genes with origins near their 3' ends, revealing that downstream origins can regulate the expression of upstream genes. This confirms previous predictions from mathematical modeling of a global causal coordination between DNA replication origin activity and mRNA expression, and shows that mathematical modeling of DNA microarray data can be used to correctly predict previously unknown biological modes of regulation.


title: Cell cycle

text: Cell cycle checkpoints are used by the cell to monitor and regulate the progress of the cell cycle. Checkpoints prevent cell cycle progression at specific points, allowing verification of necessary phase processes and repair of DNA damage. The cell cannot proceed to the next phase until checkpoint requirements have been met. Checkpoints typically consist of a network of regulatory proteins that monitor and dictate the progression of the cell through the different stages of the cell cycle.


title: Cell cycle

text: It is estimated that in normal human cells about 1% of single-strand DNA damages are converted to about 50 endogenous DNA double-strand breaks per cell per cell cycle. Although such double-strand breaks are usually repaired with high fidelity, errors in their repair are considered to contribute significantly to the rate of cancer in humans.


title: Cell cycle

text: There are several checkpoints to ensure that damaged or incomplete DNA is not passed on to daughter cells. Three main checkpoints exist: the G/S checkpoint, the G/M checkpoint and the metaphase (mitotic) checkpoint. Another checkpoint is the Go checkpoint, in which the cells are checked for maturity. If the cells fail to pass this checkpoint by not being ready yet, they will be discarded from dividing.


title: Cell cycle

text: G/S transition is a rate-limiting step in the cell cycle and is also known as restriction point. This is where the cell checks whether it has enough raw materials to fully replicate its DNA (nucleotide bases, DNA synthase, chromatin, etc.). An unhealthy or malnourished cell will get stuck at this checkpoint.


title: Cell cycle

text: The G/M checkpoint is where the cell ensures that it has enough cytoplasm and phospholipids for two daughter cells. But sometimes more importantly, it checks to see if it is the right time to replicate. There are some situations where many cells need to all replicate simultaneously (for example, a growing embryo should have a symmetric cell distribution until it reaches the mid-blastula transition). This is done by controlling the G/M checkpoint.


title: Cell cycle

text: The metaphase checkpoint is a fairly minor checkpoint, in that once a cell is in metaphase, it has committed to undergoing mitosis. However that's not to say it isn't important. In this checkpoint, the cell checks to ensure that the spindle has formed and that all of the chromosomes are aligned at the spindle equator before anaphase begins.


title: Cell cycle

text: While these are the three "main" checkpoints, not all cells have to pass through each of these checkpoints in this order to replicate. Many types of cancer are caused by mutations that allow the cells to speed through the various checkpoints or even skip them altogether. Going from S to M to S phase almost consecutively. Because these cells have lost their checkpoints, any DNA mutations that may have occurred are disregarded and passed on to the daughter cells. This is one reason why cancer cells have a tendency to exponentially accrue mutations. Aside from cancer cells, many fully differentiated cell types no longer replicate so they leave the cell cycle and stay in G until their death. Thus removing the need for cellular checkpoints. An alternative model of the cell cycle response to DNA damage has also been proposed, known as the postreplication checkpoint.


title: Cell cycle

text: Checkpoint regulation plays an important role in an organism's development. In sexual reproduction, when egg fertilization occurs, when the sperm binds to the egg, it releases signalling factors that notify the egg that it has been fertilized. Among other things, this induces the now fertilized oocyte to return from its previously dormant, G, state back into the cell cycle and on to mitotic replication and division.


title: Cell cycle

text: p53 plays an important role in triggering the control mechanisms at both G/S and G/M checkpoints. In addition to p53, checkpoint regulators are being heavily researched for their roles in cancer growth and proliferation.


title: Cell cycle

text: Pioneering work by Atsushi Miyawaki and coworkers developed the fluorescent ubiquitination-based cell cycle indicator (FUCCI), which enables fluorescence imaging of the cell cycle. Originally, a green fluorescent protein, mAG, was fused to hGem(1/110) and an orange fluorescent protein (mKO) was fused to hCdt1(30/120). Note, these fusions are fragments that contain a nuclear localization signal and ubiquitination sites for degradation, but are not functional proteins. The green fluorescent protein is made during the S, G, or M phase and degraded during the G or G phase, while the orange fluorescent protein is made during the G or G phase and destroyed during the S, G, or M phase. A far-red and near-infrared FUCCI was developed using a cyanobacteria-derived fluorescent protein (smURFP) and a bacteriophytochrome-derived fluorescent protein (movie found at this link).


title: Cell cycle

text: A disregulation of the cell cycle components may lead to tumor formation. As mentioned above, when some genes like the cell cycle inhibitors, RB, p53 etc. mutate, they may cause the cell to multiply uncontrollably, forming a tumor. Although the duration of cell cycle in tumor cells is equal to or longer than that of normal cell cycle, the proportion of cells that are in active cell division (versus quiescent cells in G phase) in tumors is much higher than that in normal tissue. Thus there is a net increase in cell number as the number of cells that die by apoptosis or senescence remains the same.


title: Cell cycle

text: The cells which are actively undergoing cell cycle are targeted in cancer therapy as the DNA is relatively exposed during cell division and hence susceptible to damage by drugs or radiation. This fact is made use of in cancer treatment; by a process known as debulking, a significant mass of the tumor is removed which pushes a significant number of the remaining tumor cells from G to G phase (due to increased availability of nutrients, oxygen, growth factors etc.). Radiation or chemotherapy following the debulking procedure kills these cells which have newly entered the cell cycle.


title: Cell cycle

text: The fastest cycling mammalian cells in culture, crypt cells in the intestinal epithelium, have a cycle time as short as 9 to 10 hours. Stem cells in resting mouse skin may have a cycle time of more than 200 hours. Most of this difference is due to the varying length of G, the most variable phase of the cycle. M and S do not vary much.


title: Cell cycle

text: For cells with a longer cell cycle time and a significantly long G phase, there is a second peak of resistance late in G.


title: Cell cycle

text: The pattern of resistance and sensitivity correlates with the level of sulfhydryl compounds in the cell. Sulfhydryls are natural substances that protect cells from radiation damage and tend to be at their highest levels in S and at their lowest near mitosis.


title: Cell cycle

text: Homologous recombination (HR) is an accurate process for repairing DNA double-strand breaks. HR is nearly absent in G1 phase, is most active in S phase, and declines in G/M. Non-homologous end joining, a less accurate and more mutagenic process for repairing double strand breaks, is active throughout the cell cycle.


title: Tupolev PAK DA

text: The Tupolev PAK DA or PAK DA (, ), codename Poslannik (), is a next-generation stealth strategic bomber being developed by Tupolev for the Long-Range Aviation branch of the Russian Aerospace Forces. The PAK DA is set to complement and eventually replace the older Tupolev Tu-95 in Russia's Air Force service.


title: Tupolev PAK DA

text: According to the 2020 Izvestia report, three PAK DA prototypes are expected to be ready for preliminary testing by April 2023, with the state tests to begin in February 2026. The aircraft is expected to enter serial production in 2027. Due to the ongoing modernization of Russia's current bomber fleet and massive purchases of upgraded Tu-160M2s, it is believed that the Russian Defence Ministry will initially procure only a small number of PAK DA bombers.


title: Tupolev PAK DA

text: Technical parameters of the PAK DA include subsonic speed, 12,000 km operational range and a capability to continuously remain in the air for up to 30 hours while carrying both conventional and nuclear payloads up to 30 tons. The aircraft is expected to have a crew of 4.


title: Tupolev PAK DA

text: The first mention of Russia's next-generation long-range strategic bomber dates back to the late 1990s when the formation of requirements for the aircraft had begun. In December 2007, the Russian Air Force handed to Tupolev Design Bureau the first set of technical and tactical requirements for a new strategic bomber and financing of the programme began in 2008. According to some early reports, the PAK DA was to be heavily based on the supersonic Tupolev Tu-160, but later reports regarding to the aircraft, including a televised address from then-Prime Minister Vladimir Putin, seemed to imply that it will be an entirely new design.


title: Tupolev PAK DA

text: On 3 September 2009, the Russian Defence Ministry awarded Tupolev Design Bureau a three-year R&D contract to undertake studies for the new long-range bomber. According to the President-General Designer of Tupolev, Igor Shevchuk, "this should be a fundamentally new aircraft, based on conceptually new solutions".


title: Tupolev PAK DA

text: In June 2012, then-Deputy Prime Minister Dmitry Rogozin questioned the need for a new long-range bomber, pointing out the great progress in air defence and anti-missile defence technology, saying "these aircraft will not get anywhere. Not ours, not theirs." The Chief of the General Staff Nikolai Makarov, responded by stating that work is ongoing and that the design was superior to American aircraft. On 9 June 2012, Prime Minister Dmitry Medvedev stated that the PAK DA was confirmed as planned. Russian President Vladimir Putin in his statement on 14 June 2012 also urged the need for a new long-range bomber.


title: Tupolev PAK DA

text: In March 2013, it was reported that the selected PAK DA design would be a subsonic flying wing with emphasis on stealth technology rather than on a capability to overcome supersonic speeds.


title: Tupolev PAK DA

text: On 30 August 2013, a Russian Defense Ministry source revealed that the PAK DA will be equipped with advanced types of precision-guided weapons, including hypersonic weapons. The bomber itself will fly at subsonic speeds.


title: Tupolev PAK DA

text: In November 2013, a decision was taken to speed up work on the PAK DA and begin full-scale R&D work in 2014.


title: Tupolev PAK DA

text: In February 2014, then-Deputy Minister of Industry and Trade Andrey Boginsky stated that Russia was attempting to acquire Chinese investment in the project.


title: Tupolev PAK DA

text: In April 2014, head of Russia's United Aircraft Corporation (UAC) Mikhail Pogosyan announced that the Tupolev Design Bureau had finalized the design of the PAK DA and that the project was moving to the intermediate phase - i.e. completion of the design and construction of a prototype.


title: Tupolev PAK DA

text: In March 2015, it became known that the KAPO plant in Kazan was chosen for the construction of the PAK DA prototypes and serial aircraft. The Russian Defence Ministry allocated about RUB5 billion for reconstruction and technical re-equipment of the production base, to prepare the plant for the construction of the new bomber. Development of the PAK DA will be carried out simultaneously with serial production of the upgraded Tu-160M2, although this later led to repeated delays in the programme.


title: Tupolev PAK DA

text: In July 2015, representatives from the United Instrument Manufacturing Corporation (UIMC), now part of Roselectronics, announced the company is developing "one-of-a-kind communications system" for the bomber.


title: Tupolev PAK DA

text: On 1 March 2017, it was reported that the first full-size model of the bomber was built, amongst various scale mock-ups of different configurations.


title: Tupolev PAK DA

text: In July 2018, United Engine Corporation (UEC) and Tupolev signed a contract for the creation of PAK DA's powerplant. According to early reports, it will be derived from the upgraded Kuznetsov NK-32 Tier 2 engine, intended for modernized Tu-160M2s. About RUB8 billion was to be allocated for the development of the new engine that will be produced by JSC Kuznetsov in Samara. The engine is to provide the PAK DA with a capability to perform 30 hours nonstop flight while being resistant to temperatures from minus 60°C to plus 50°C and even effects of a nuclear explosion. Its service life is expected to be from 12 to 21 years. The first tests of the PAK DA engine will take place in 2020 and be completed by the end of 2021.


title: Tupolev PAK DA

text: In February 2019, PAK DA's final draft was approved and all documents for the construction of the bomber were signed. Construction of the first aircraft components began in late 2019.


title: Tupolev PAK DA

text: On 26 May 2020, the beginning of construction of the first "PAK DA" bomber was reported by TASS with reference to anonymous sources within the industry. Still according to TASS, citing another source in the aircraft industry, the first prototype of the PAK DA will be finished in early 2023.


title: Tupolev PAK DA

text: On 23 December 2020, it was reported that United Engine Corporation was in the final stages of completing the first prototype engine for the PAK DA, with bench tests to be carried out in 2021. The new engine was designed digitally, which made it possible to quickly manufacture the first sample. On 24 December 2020, according to a TASS source in the Russian military-industrial complex, it was reported that there were now several PAK DA being assembled at the same time at the Kazan aircraft plant. The source said that two or three prototypes are already being launched into production.


title: Tupolev PAK DA

text: On 22 July 2021, the general director of NPP Zvezda announced at the 2021 MAKS air show that the ejection seats for the PAK DA were under development and should be handed over to Tupolev in 2023. The testing phase of the PAK DA's ejection seat started in October 2022, according to the general director of NPP Zvezda.


title: Tupolev PAK DA

text: On 2 August 2021, it was announced that a new long-range nuclear-capable hypersonic cruise missile, the Kh-95, was under development, to be fitted inside the PAK DA's weapon bay.


title: Tupolev PAK DA

text: On 25 November 2021, it was reported, according to a source in the Russian military-industrial complex, that the first PAK DA prototype was in final stage of assembly and would start ground tests in 2022, with the first flight scheduled for late 2024 or early 2025.


title: Tupolev PAK DA

text: On 16 March 2022, Tupolev published a patent for an engine air intake for stealthy flying wing aircraft, including illustrations of such an aircraft.


title: Tupolev PAK DA

text: On 13 May 2022, a production planning of United Aircraft Corporation at the Voronezh Aircraft Production Association plant showed that the production of components for 6 PAK DA was ongoing. The planning also confirmed that several PAK DA prototypes were still expected to be built between 2023 and 2026. Work on the schedule was further warranted by Russian Deputy Prime Minister Denis Manturov, explaining that the sanctions following the 2022 Russian invasion of Ukraine would not affect the timing of the creation of the PAK DA as the bomber does not contain any foreign components.


title: Continuous function

text: In mathematics, a continuous function is a function such that a continuous variation (that is a change without jump) of the argument induces a continuous variation of the value of the function. This means that there are no abrupt changes in value, known as "discontinuities". More precisely, a function is continuous if arbitrarily small changes in its value can be assured by restricting to sufficiently small changes of its argument. A discontinuous function is a function that is . Up until the 19th century, mathematicians largely relied on intuitive notions of continuity, and considered only continuous functions. The epsilon–delta definition of a limit was introduced to formalize the definition of continuity.


title: Continuous function

text: Continuity is one of the core concepts of calculus and mathematical analysis, where arguments and values of functions are real and complex numbers. The concept has been generalized to functions between metric spaces and between topological spaces. The latter are the most general continuous functions, and their definition is the basis of topology.


title: Continuous function

text: A stronger form of continuity is uniform continuity. In order theory, especially in domain theory, a related concept of continuity is Scott continuity.


title: Continuous function

text: As an example, the function denoting the height of a growing flower at time would be considered continuous. In contrast, the function denoting the amount of money in a bank account at time would be considered discontinuous, since it "jumps" at each point in time when money is deposited or withdrawn.


title: Continuous function

text: A form of the epsilon–delta definition of continuity was first given by Bernard Bolzano in 1817. Augustin-Louis Cauchy defined continuity of formula_1 as follows: an infinitely small increment formula_2 of the independent variable "x" always produces an infinitely small change formula_3 of the dependent variable "y" (see e.g. "Cours d'Analyse", p. 34). Cauchy defined infinitely small quantities in terms of variable quantities, and his definition of continuity closely parallels the infinitesimal definition used today (see microcontinuity). The formal definition and the distinction between pointwise continuity and uniform continuity were first given by Bolzano in the 1830s but the work wasn't published until the 1930s. Like Bolzano, Karl Weierstrass denied continuity of a function at a point "c" unless it was defined at and on both sides of "c", but Édouard Goursat allowed the function to be defined only at and on one side of "c", and Camille Jordan allowed it even if the function was defined only at "c". All three of those nonequivalent definitions of pointwise continuity are still in use. Eduard Heine provided the first published definition of uniform continuity in 1872, but based these ideas on lectures given by Peter Gustav Lejeune Dirichlet in 1854.


title: Continuous function

text: A real function, that is a function from real numbers to real numbers, can be represented by a graph in the Cartesian plane; such a function is continuous if, roughly speaking, the graph is a single unbroken curve whose domain is the entire real line. A more mathematically rigorous definition is given below.


title: Continuous function

text: Continuity of real functions is usually defined in terms of limits. A function with variable is "continuous at" the real number , if the limit of formula_4 as tends to , is equal to formula_5


title: Continuous function

text: There are several different definitions of (global) continuity of a function, which depend on the nature of its domain.


title: Continuous function

text: A function is continuous on an open interval if the interval is contained in the domain of the function, and the function is continuous at every point of the interval. A function that is continuous on the interval formula_6 (the whole real line) is often called simply a continuous function; one says also that such a function is "continuous everywhere". For example, all polynomial functions are continuous everywhere.


title: Continuous function

text: A function is continuous on a semi-open or a closed interval, if the interval is contained in the domain of the function, the function is continuous at every interior point of the interval, and the value of the function at each endpoint that belongs to the interval is the limit of the values of the function when the variable tends to the endpoint from the interior of the interval. For example, the function formula_7 is continuous on its whole domain, which is the closed interval formula_8


title: Continuous function

text: Many commonly encountered functions are partial functions that have a domain formed by all real numbers, except some isolated points. Examples are the functions formula_9 and formula_10 When they are continuous on their domain, one says, in some contexts, that they are continuous, although they are not continuous everywhere. In other contexts, mainly when one is interested with their behavior near the exceptional points, one says that they are discontinuous.


title: Continuous function

text: A partial function is "discontinuous" at a point, if the point belongs to the topological closure of its domain, and either the point does not belong to the domain of the function, or the function is not continuous at the point. For example, the functions formula_11 and formula_12 are discontinuous at , and remain discontinuous whichever value is chosen for defining them at . A point where a function is discontinuous is called a "discontinuity".


title: Continuous function

text: Using mathematical notation, there are several ways to define continuous functions in each of the three senses mentioned above.


title: Continuous function

text: In case of the domain formula_14 being defined as an open interval, formula_24 and formula_25 do not belong to formula_14, and the values of formula_27 and formula_28 do not matter for continuity on formula_14.


title: Continuous function

text: The function is "continuous at some point" of its domain if the limit of formula_4 as "x" approaches "c" through the domain of "f", exists and is equal to formula_5 In mathematical notation, this is written as


title: Continuous function

text: In detail this means three conditions: first, has to be defined at (guaranteed by the requirement that is in the domain of ). Second, the limit on the left hand side of that equation has to exist. Third, the value of this limit must equal formula_5


title: Continuous function

text: A neighborhood of a point "c" is a set that contains, at least, all points within some fixed distance of "c". Intuitively, a function is continuous at a point "c" if the range of "f" over the neighborhood of "c" shrinks to a single point formula_34 as the width of the neighborhood around "c" shrinks to zero. More precisely, a function "f" is continuous at a point "c" of its domain if, for any neighborhood formula_35 there is a neighborhood formula_36 in its domain such that formula_37 whenever formula_38


title: Continuous function

text: This definition only requires that the domain and the codomain are topological spaces and is thus the most general definition. It follows from this definition that a function "f" is automatically continuous at every isolated point of its domain. As a specific example, every real valued function on the set of integers is continuous.


title: Continuous function

text: One can instead require that for any sequence formula_39 of points in the domain which converges to "c", the corresponding sequence formula_40 converges to formula_5 In mathematical notation, formula_42


title: Continuous function

text: Explicitly including the definition of the limit of a function, we obtain a self-contained definition: Given a function formula_43 as above and an element formula_44 of the domain formula_14, formula_46 is said to be continuous at the point formula_44 when the following holds: For any positive real number formula_48 however small, there exists some positive real number formula_49 such that for all formula_50 in the domain of formula_46 with formula_52 the value of formula_53 satisfies


title: Continuous function

text: Alternatively written, continuity of formula_43 at formula_56 means that for every formula_48 there exists a formula_49 such that for all formula_59:


title: Continuous function

text: More intuitively, we can say that if we want to get all the formula_53 values to stay in some small neighborhood around formula_62 we simply need to choose a small enough neighborhood for the formula_50 values around formula_64 If we can do that no matter how small the formula_65 neighborhood is, then formula_46 is continuous at formula_64


title: Continuous function

text: In modern terms, this is generalized by the definition of continuity of a function with respect to a basis for the topology, here the metric topology.


title: Continuous function

text: Weierstrass had required that the interval formula_68 be entirely within the domain formula_14, but Jordan removed that restriction.


title: Continuous function

text: In proofs and numerical analysis we often need to know how fast limits are converging, or in other words, control of the remainder. We can formalize this to a definition of continuity.


title: Continuous function

text: A function formula_72 is "C"-continuous at formula_44 if there exists such a neighbourhood formula_74 that


title: Continuous function

text: This approach leads naturally to refining the notion of continuity by restricting the set of admissible control functions. For a given set of control functions formula_77 a function is if it is for some formula_78 For example, the Lipschitz and Hölder continuous functions of exponent below are defined by the set of control functions


title: Continuous function

text: Continuity can also be defined in terms of oscillation: a function "f" is continuous at a point formula_44 if and only if its oscillation at that point is zero; in symbols, formula_82 A benefit of this definition is that it discontinuity: the oscillation gives how the function is discontinuous at a point.


title: Continuous function

text: This definition is useful in descriptive set theory to study the set of discontinuities and continuous points – the continuous points are the intersection of the sets where the oscillation is less than formula_83 (hence a formula_84 set) – and gives a very quick proof of one direction of the Lebesgue integrability condition.


title: Continuous function

text: The oscillation is equivalent to the formula_85 definition by a simple re-arrangement, and by using a limit (lim sup, lim inf) to define oscillation: if (at a given point) for a given formula_86 there is no formula_87 that satisfies the formula_85 definition, then the oscillation is at least formula_89 and conversely if for every formula_83 there is a desired formula_91 the oscillation is 0. The oscillation definition can be naturally generalized to maps from a topological space to a metric space.


title: Continuous function

text: Cauchy defined continuity of a function in the following intuitive terms: an infinitesimal change in the independent variable corresponds to an infinitesimal change of the dependent variable (see "Cours d'analyse", page 34). Non-standard analysis is a way of making this mathematically rigorous. The real line is augmented by the addition of infinite and infinitesimal numbers to form the hyperreal numbers. In nonstandard analysis, continuity can be defined as follows.


title: Continuous function

text: (see microcontinuity). In other words, an infinitesimal increment of the independent variable always produces to an infinitesimal change of the dependent variable, giving a modern expression to Augustin-Louis Cauchy's definition of continuity.


title: Continuous function

text: Checking the continuity of a given function can be simplified by checking one of the above defining properties for the building blocks of the given function. It is straightforward to show that the sum of two functions, continuous on some domain, is also continuous on this domain. Given


title: Continuous function

text: Combining the above preservations of continuity and the continuity of constant functions and of the identity function formula_101 one arrives at the continuity of all polynomial functions such as


title: Continuous function

text: is defined for all real numbers formula_115 and is continuous at every such point. Thus it is a continuous function. The question of continuity at formula_116 does not arise, since formula_116 is not in the domain of formula_118 There is no continuous function formula_119 that agrees with formula_120 for all formula_121


title: Continuous function

text: Since the function sine is continuous on all reals, the sinc function formula_122 is defined and continuous for all real formula_123 However, unlike the previous example, "G" be extended to a continuous function on real numbers, by the value formula_124 to be 1, which is the limit of formula_125 when "x" approaches 0, i.e.,


title: Continuous function

text: the sinc-function becomes a continuous function on all real numbers. The term is used in such cases, when (re)defining values of a function to coincide with the appropriate limits make a function continuous at specific points.


title: Continuous function

text: A more involved construction of continuous functions is the function composition. Given two continuous functions


title: Continuous function

text: Pick for instance formula_135. Then there is no around formula_136, i.e. no open interval formula_137 with formula_138 that will force all the formula_139 values to be within the of formula_140, i.e. within formula_141. Intuitively we can think of this type of discontinuity as a sudden jump in function values.


title: Continuous function

text: Besides plausible continuities and discontinuities like above, there are also functions with a behavior, often coined pathological, for example, Thomae's function,


title: Continuous function

text: is continuous at all irrational numbers and discontinuous at all rational numbers. In a similar vein, Dirichlet's function, the indicator function for the set of rational numbers,


title: Continuous function

text: Let formula_53 be a function that is continuous at a point formula_149 and formula_150 be a value such formula_151 Then formula_152 throughout some neighbourhood of formula_64


title: Continuous function

text: Suppose there is a point in the neighbourhood formula_157 for which formula_158 then we have the contradiction


title: Continuous function

text: The intermediate value theorem is an existence theorem, based on the real number property of completeness, and states:


title: Continuous function

text: For example, if a child grows from 1 m to 1.5 m between the ages of two and six years, then, at some time between two and six years of age, the child's height must have been 1.25 m.


title: Continuous function

text: As a consequence, if "f" is continuous on formula_165 and formula_27 and formula_28 differ in sign, then, at some point formula_163 formula_34 must equal zero.


title: Continuous function

text: The extreme value theorem states that if a function "f" is defined on a closed interval formula_165 (or any closed and bounded set) and is continuous there, then the function attains its maximum, i.e. there exists formula_171 with formula_172 for all formula_173 The same is true of the minimum of "f". These statements are not, in general, true if the function is defined on an open interval formula_174 (or any set that is not both closed and bounded), as, for example, the continuous function formula_175 defined on the open interval (0,1), does not attain a maximum, being unbounded above.


title: Continuous function

text: is everywhere continuous. However, it is not differentiable at formula_136 (but is so everywhere else). Weierstrass's function is also everywhere continuous but nowhere differentiable.


title: Continuous function

text: The derivative "f′"("x") of a differentiable function "f"("x") need not be continuous. If "f′"("x") is continuous, "f"("x") is said to be "continuously differentiable". The set of such functions is denoted formula_179 More generally, the set of functions


title: Continuous function

text: (from an open interval (or open subset of formula_15) formula_182 to the reals) such that "f" is formula_183 times differentiable and such that the formula_183-th derivative of "f" is continuous is denoted formula_185 See differentiability class. In the field of computer graphics, properties related (but not identical) to formula_186 are sometimes called formula_187 (continuity of position), formula_188 (continuity of tangency), and formula_189 (continuity of curvature); see Smoothness of curves and surfaces.


title: Continuous function

text: is integrable (for example in the sense of the Riemann integral). The converse does not hold, as the (integrable, but discontinuous) sign function shows.


title: Continuous function

text: exists for all formula_193, the resulting function formula_53 is referred to as the pointwise limit of the sequence of functions formula_195 The pointwise limit function need not be continuous, even if all functions formula_196 are continuous, as the animation at the right shows. However, "f" is continuous if all functions formula_196 are continuous and the sequence converges uniformly, by the uniform convergence theorem. This theorem can be used to show that the exponential functions, logarithms, square root function, and trigonometric functions are continuous.


title: Continuous function

text: Discontinuous functions may be discontinuous in a restricted way, giving rise to the concept of directional continuity (or right and left continuous functions) and semi-continuity. Roughly speaking, a function is if no jump occurs when the limit point is approached from the right. Formally, "f" is said to be right-continuous at the point "c" if the following holds: For any number formula_198 however small, there exists some number formula_49 such that for all "x" in the domain with formula_200 the value of formula_53 will satisfy


title: Continuous function

text: This is the same condition as for continuous functions, except that it is required to hold for "x" strictly larger than "c" only. Requiring it instead for all "x" with formula_203 yields the notion of functions. A function is continuous if and only if it is both right-continuous and left-continuous.


title: Continuous function

text: A function "f" is if, roughly, any jumps that might occur only go down, but not up. That is, for any formula_48 there exists some number formula_49 such that for all "x" in the domain with formula_206 the value of formula_53 satisfies


title: Continuous function

text: The concept of continuous real-valued functions can be generalized to functions between metric spaces. A metric space is a set formula_209 equipped with a function (called metric) formula_210 that can be thought of as a measurement of the distance of any two elements in "X". Formally, the metric is a function


title: Continuous function

text: that satisfies a number of requirements, notably the triangle inequality. Given two metric spaces formula_212 and formula_213 and a function


title: Continuous function

text: then formula_46 is continuous at the point formula_216 (with respect to the given metrics) if for any positive real number formula_48 there exists a positive real number formula_49 such that all formula_219 satisfying formula_220 will also satisfy formula_221 As in the case of real functions above, this is equivalent to the condition that for every sequence formula_222 in formula_209 with limit formula_224 we have formula_225 The latter condition can be weakened as follows: formula_46 is continuous at the point formula_227 if and only if for every convergent sequence formula_222 in formula_209 with limit formula_227, the sequence formula_231 is a Cauchy sequence, and formula_227 is in the domain of formula_46.


title: Continuous function

text: The set of points at which a function between metric spaces is continuous is a formula_84 set – this follows from the formula_85 definition of continuity.


title: Continuous function

text: This notion of continuity is applied, for example, in functional analysis. A key statement in this area says that a linear operator


title: Continuous function

text: between normed vector spaces formula_237 and formula_238 (which are vector spaces equipped with a compatible norm, denoted formula_239) is continuous if and only if it is bounded, that is, there is a constant formula_240 such that


title: Continuous function

text: The concept of continuity for functions between metric spaces can be strengthened in various ways by limiting the way formula_87 depends on formula_83 and "c" in the definition above. Intuitively, a function "f" as above is uniformly continuous if the formula_87 does


title: Continuous function

text: not depend on the point "c". More precisely, it is required that for every real number formula_198 there exists formula_49 such that for every formula_248 with formula_249 we have that formula_250 Thus, any uniformly continuous function is continuous. The converse does not hold in general, but holds when the domain space "X" is compact. Uniformly continuous maps can be defined in the more general situation of uniform spaces.


title: Continuous function

text: A function is Hölder continuous with exponent α (a real number) if there is a constant "K" such that for all formula_251 the inequality


title: Continuous function

text: holds. Any Hölder continuous function is uniformly continuous. The particular case formula_253 is referred to as Lipschitz continuity. That is, a function is Lipschitz continuous if there is a constant "K" such that the inequality


title: Continuous function

text: holds for any formula_255 The Lipschitz condition occurs, for example, in the Picard–Lindelöf theorem concerning the solutions of ordinary differential equations.


title: Continuous function

text: Another, more abstract, notion of continuity is continuity of functions between topological spaces in which there generally is no formal notion of distance, as there is in the case of metric spaces. A topological space is a set "X" together with a topology on "X", which is a set of subsets of "X" satisfying a few requirements with respect to their unions and intersections that generalize the properties of the open balls in metric spaces while still allowing to talk about the neighbourhoods of a given point. The elements of a topology are called open subsets of "X" (with respect to the topology).


title: Continuous function

text: between two topological spaces "X" and "Y" is continuous if for every open set formula_257 the inverse image


title: Continuous function

text: is an open subset of "X". That is, "f" is a function between the sets "X" and "Y" (not on the elements of the topology formula_259), but the continuity of "f" depends on the topologies used on "X" and "Y".