Projectively extended real line

In real analysis, the projectively extended real line (also called the one-point compactification of the real line), is the extension of the set of the real numbers, formula_1 by a point denoted . It is thus the set formula_2 with the standard arithmetic operations extended where possible, and is sometimes denoted by formula_3 The added point is called the point at infinity, because it is considered as a neighbour of both ends of the real line. More precisely, the point at infinity is the limit of every sequence of real numbers whose absolute values are increasing and unbounded.
The projectively extended real line may be identified with the projective line over the reals in which three points have been assigned specific values (e.g. , and ). The projectively extended real line is distinct from the extended real number line, in which and are distinct.
Dividing by zero.
Unlike most mathematical models of the intuitive concept of 'number', this structure allows division by zero:
for nonzero "a". In particular , and moreover , making reciprocal, , a total function in this structure. The structure, however, is not a field, and none of the binary arithmetic operations are total, as witnessed for example by being undefined despite the reciprocal being total. It has usable interpretations, however – for example, in geometry, a vertical line has "infinite" slope.
Extensions of the real line.
The projectively extended real line extends the field of real numbers in the same way that the Riemann sphere extends the field of complex numbers, by adding a single point called conventionally formula_5.
In contrast, the extended real number line (also called the two-point compactification of the real line) distinguishes between formula_6 and formula_7.
Order.
The order relation cannot be extended to formula_8 in a meaningful way. Given a number formula_9, there is no convincing argument to define either formula_10 or that formula_11. Since formula_5 can't be compared with any of the other elements, there's no point in retaining this relation on formula_8. However, order on formula_14 is used in definitions in formula_8.
Geometry.
Fundamental to the idea that ∞ is a point "no different from any other" is the way the real projective line is a homogeneous space, in fact homeomorphic to a circle. For example the general linear group of 2×2 real invertible matrices has a transitive action on it. The group action may be expressed by Möbius transformations, (also called linear fractional transformations), with the understanding that when the denominator of the linear fractional transformation is 0, the image is ∞.
The detailed analysis of the action shows that for any three distinct points "P", "Q" and "R", there is a linear fractional transformation taking "P" to 0, "Q" to 1, and "R" to ∞ that is, the group of linear fractional transformations is triply transitive on the real projective line. This cannot be extended to 4-tuples of points, because the cross-ratio is invariant.
The terminology projective line is appropriate, because the points are in 1-to-1 correspondence with one-dimensional linear subspaces of formula_16.
Arithmetic operations.
Motivation for arithmetic operations.
The arithmetic operations on this space are an extension of the same operations on reals. A motivation for the new definitions is the limits of functions of real numbers.
Arithmetic operations that are defined.
In addition to the standard operations on the subset formula_14 of formula_8, the following operations are defined for formula_19, with exceptions as indicated:
Arithmetic operations that are left undefined.
The following expressions cannot be motivated by considering limits of real functions, and no definition of them allows the statement of the standard algebraic properties to be retained unchanged in form for all defined cases.</math>, resolve to the standard rules: see Wheel theory.}} Consequently, they are left undefined:
Algebraic properties.
The following equalities mean: "Either both sides are undefined, or both sides are defined and equal." This is true for any formula_22.
The following is true whenever the right-hand side is defined, for any formula_22.
In general, all laws of arithmetic that are valid for formula_14 are also valid for formula_8 whenever all the occurring expressions are defined.
Intervals and topology.
The concept of an interval can be extended to formula_8. However, since it is an unordered set, the interval has a slightly different meaning. The definitions for closed intervals are as follows (it is assumed that 
formula_29):
With the exception of when the end-points are equal, the corresponding open and half-open intervals are defined by removing the respective endpoints.
The open intervals as base define a topology on formula_8. Sufficient for a base are the finite open intervals in formula_14 and the intervals formula_40 for all formula_41 such that formula_42.
As said, the topology is homeomorphic to a circle. Thus it is metrizable corresponding (for a given homeomorphism) to the ordinary metric on this circle (either measured straight or along the circle). There is no metric which is an extension of the ordinary metric on formula_14.
Interval arithmetic.
Interval arithmetic extends to formula_8 from formula_14. The result of an arithmetic operation on intervals is always an interval, except when the intervals with a binary operation contain incompatible values leading to an undefined result. In particular, we have, for every formula_46:
irrespective of whether either interval includes formula_48 and formula_5.
Calculus.
The tools of calculus can be used to analyze functions of formula_8. The definitions are motivated by the topology of this space.
Neighbourhoods.
Let formula_51.
Limits.
Basic definitions of limits.
Let formula_59.
The limit of "f"("x") as "x" approaches "p" is "L", denoted
if and only if for every neighbourhood "A" of "L", there is a punctured neighbourhood "B" of "p", such that formula_52 implies formula_62.
The one-sided limit of "f"("x") as "x" approaches "p" from the right (left) is "L", denoted
if and only if for every neighbourhood "A" of "L", there is a right-sided (left-sided) punctured neighbourhood "B" of "p", such that formula_52 implies formula_62.
It can be shown that formula_60 if and only if both formula_67 and formula_68.
Comparison with limits in formula_14.
The definitions given above can be compared with the usual definitions of limits of real functions. In the following statements, formula_70, the first limit is as defined above, and the second limit is in the usual sense:
Extended definition of limits.
Let formula_83. Then "p" is a limit point of "A" if and only if every neighbourhood of "p" includes a point formula_84 such that formula_85.
Let formula_86, "p" a limit point of "A". The limit of "f(x)" as "x" approaches "p" through "A" is "L", if and only if for every neighbourhood "B" of "L", there is a punctured neighbourhood "C" of "p", such that formula_87 implies formula_88.
This corresponds to the regular topological definition of continuity, applied to the subspace topology on formula_89, and the restriction of "f" to formula_89.
Continuity.
The function
is continuous at if and only if is defined at and
If formula_93 the function
is continuous in if and only if, for every formula_95, is defined at and the limit of as tends to through is .
Every rational function , where and are polynomials, can be prolongated, in a unique way, to a function from formula_8 to formula_8 that is continuous in formula_8. In particular, this is the case of polynomial functions, which take the value formula_5 at formula_100 if they are not constant.
Also, if the tangent function is extended so that
then is continuous in formula_1 but cannot be prolongated further to a function that is continuous in formula_3
Many elementary functions that are continuous in formula_104 cannot be prolongated to functions that are continuous in formula_105 This is the case, for example, of the exponential function and all trigonometric functions. For example, the sine function is continuous in formula_1 but it cannot be made continuous at formula_107 As seen above, the tangent function can be prolongated to a function that is continuous in formula_1 but this function cannot be made continuous at formula_107
Many discontinuous functions that become continuous when the codomain is extended to formula_8 remain discontinuous if the codomain is extended to the affinely extended real number system formula_111 This is the case of the function formula_112 On the other hand, some functions that are continuous in formula_104 and discontinuous at formula_114 become continuous if the domain is extended to formula_111 This is the case of the arc tangent.
As a projective range.
When the real projective line is considered in the context of the real projective plane, then the consequences of Desargues' theorem are implicit. In particular, the construction of the projective harmonic conjugate relation between points is part of the structure of the real projective line. For instance, given any pair of points, the point at infinity is the projective harmonic conjugate of their midpoint.
As projectivities preserve the harmonic relation, they form the automorphisms of the real projective line. The projectivities are described algebraically as homographies, since the real numbers form a ring, according to the general construction of a projective line over a ring. Collectively they form the group PGL(2,R).
The projectivities which are their own inverses are called involutions. A hyperbolic involution has two fixed points. Two of these correspond to elementary, arithmetic operations on the real projective line: negation and reciprocation. Indeed, 0 and ∞ are fixed under negation, while 1 and −1 are fixed under reciprocation.