Lipschitz continuity

In mathematical analysis, Lipschitz continuity, named after Rudolf Lipschitz, is a strong form of uniform continuity for functions. Intuitively, a Lipschitz continuous function is limited in how fast it can change: there exists a real number such that, for every pair of points on the graph of this function, the absolute value of the slope of the line connecting them is not greater than this real number; the smallest such bound is called the "Lipschitz constant" of the function (or "modulus of uniform continuity"). For instance, every function that has bounded first derivatives is Lipschitz continuous.
In the theory of differential equations, Lipschitz continuity is the central condition of the Picard–Lindelöf theorem which guarantees the existence and uniqueness of the solution to an initial value problem. A special type of Lipschitz continuity, called contraction, is used in the Banach fixed-point theorem.
We have the following chain of strict inclusions for functions over a closed and bounded non-trivial interval of the real line
where 0 < α ≤ 1. We also have
Definitions.
Given two metric spaces ("X", "d""X") and ("Y", "d""Y"), where "d""X" denotes the metric on the set "X" and "d""Y" is the metric on set "Y", a function "f" : "X" → "Y" is called Lipschitz continuous if there exists a real constant "K" ≥ 0 such that, for all "x"1 and "x"2 in "X",
Any such "K" is referred to as a Lipschitz constant for the function "f". The smallest constant is sometimes called the (best) Lipschitz constant; however, in most cases, the latter notion is less relevant. If "K" = 1 the function is called a short map, and if 0 ≤ "K" < 1 and "f" maps a metric space to itself, the function is called a contraction.
In particular, a real-valued function "f" : "R" → "R" is called Lipschitz continuous if there exists a positive real constant K such that, for all real "x"1 and "x"2,
In this case, "Y" is the set of real numbers R with the standard metric "d""Y"("y1", "y2") = |"y1" − "y2"|, and "X" is a subset of R.
In general, the inequality is (trivially) satisfied if "x"1 = "x"2. Otherwise, one can equivalently define a function to be Lipschitz continuous if and only if there exists a constant "K" ≥ 0 such that, for all "x"1 ≠ "x"2, 
For real-valued functions of several real variables, this holds if and only if the absolute value of the slopes of all secant lines are bounded by "K". The set of lines of slope "K" passing through a point on the graph of the function forms a circular cone, and a function is Lipschitz if and only if the graph of the function everywhere lies completely outside of this cone (see figure).
A function is called locally Lipschitz continuous if for every "x" in "X" there exists a neighborhood "U" of "x" such that "f" restricted to "U" is Lipschitz continuous. Equivalently, if "X" is a locally compact metric space, then "f" is locally Lipschitz if and only if it is Lipschitz continuous on every compact subset of "X". In spaces that are not locally compact, this is a necessary but not a sufficient condition.
More generally, a function "f" defined on "X" is said to be Hölder continuous or to satisfy a Hölder condition of order α > 0 on "X" if there exists a constant "M" ≥ 0 such that
for all "x" and "y" in "X". Sometimes a Hölder condition of order α is also called a uniform Lipschitz condition of order α > 0.
If there exists a "K" ≥ 1 with
then "f" is called bilipschitz (also written bi-Lipschitz). A bilipschitz mapping is injective, and is in fact a homeomorphism onto its image. A bilipschitz function is the same thing as an injective Lipschitz function whose inverse function is also Lipschitz.
Lipschitz manifolds.
Let "U" and "V" be two open sets in R"n". A function "T" : "U" → "V" is called bi-Lipschitz if it is a Lipschitz homeomorphism onto its image, and its inverse is also Lipschitz.
Using bi-Lipschitz mappings, it is possible to define a Lipschitz structure on a topological manifold, since there is a pseudogroup structure on bi-Lipschitz homeomorphisms. This structure is intermediate between that of a piecewise-linear manifold and a smooth manifold. In fact a PL structure gives rise to a unique Lipschitz structure; it can in that sense 'nearly' be smoothed.
One-sided Lipschitz.
Let "F"("x") be an upper semi-continuous function of "x", and that "F"("x") is a closed, convex set for all "x". Then "F" is one-sided Lipschitz if
for some "C" and for all "x"1 and "x"2.
It is possible that the function "F" could have a very large Lipschitz constant but a moderately sized, or even negative, one-sided Lipschitz constant. For example, the function
has Lipschitz constant "K" = 50 and a one-sided Lipschitz constant "C" = 0. An example which is one-sided Lipschitz but not Lipschitz continuous is "F"("x") = "e"−"x", with "C" = 0.