Extended real number line

"Positive infinity" redirects here. For the band, see Positive Infinity.

In mathematics, the affinely extended real number system is obtained from the real number system by adding two elements: + and (read as positive infinity and negative infinity respectively). These new elements are not real numbers. It is useful in describing various limiting behaviors in calculus and mathematical analysis, especially in the theory of measure and integration. The affinely extended real number system is denoted or [–∞, +∞] or ℝ ∪ {–∞, +∞}.

When the meaning is clear from context, the symbol +∞ is often written simply as .

Motivation

Limits

We often wish to describe the behavior of a function , as either the argument or the function value gets "very big" in some sense. For example, consider the function

The graph of this function has a horizontal asymptote at y = 0. Geometrically, as we move farther and farther to the right along the -axis, the value of approaches 0. This limiting behavior is similar to the limit of a function at a real number, except that there is no real number to which approaches.

By adjoining the elements and to , we allow a formulation of a "limit at infinity" with topological properties similar to those for .

To make things completely formal, the Cauchy sequences definition of allows us to define as the set of all sequences of rationals which, for any , from some point on exceed . We can define similarly.

Measure and integration

In measure theory, it is often useful to allow sets which have infinite measure and integrals whose value may be infinite.

Such measures arise naturally out of calculus. For example, in assigning a measure to that agrees with the usual length of intervals, this measure must be larger than any finite real number. Also, when considering infinite integrals, such as

the value "infinity" arises. Finally, it is often useful to consider the limit of a sequence of functions, such as

Without allowing functions to take on infinite values, such essential results as the monotone convergence theorem and the dominated convergence theorem would not make sense.

Order and topological properties

The affinely extended real number system turns into a totally ordered set by defining for all . This order has the desirable property that every subset has a supremum and an infimum: it is a complete lattice.

This induces the order topology on . In this topology, a set is a neighborhood of if and only if it contains a set for some real number , and analogously for the neighborhoods of . is a compact Hausdorff space homeomorphic to the unit interval . Thus the topology is metrizable, corresponding (for a given homeomorphism) to the ordinary metric on this interval. There is no metric which is an extension of the ordinary metric on .

With this topology the specially defined limits for tending to and , and the specially defined concepts of limits equal to and , reduce to the general topological definitions of limits.

Arithmetic operations

The arithmetic operations of can be partially extended to as follows:

For exponentiation, see Exponentiation#Limits of powers. Here, "" means both "" and "", while "" means both "" and "".

The expressions and (called indeterminate forms) are usually left undefined. These rules are modeled on the laws for infinite limits. However, in the context of probability or measure theory, is often defined as .

The expression is not defined either as or , because although it is true that whenever for a continuous function it must be the case that is eventually contained in every neighborhood of the set , it is not true that must tend to one of these points. An example is (as goes to infinity). (The modulus , nevertheless, does approach .)

Algebraic properties

With these definitions is not even a semigroup, let alone a group, a ring or a field, like is one. However, it still has several convenient properties:

• and are either equal or both undefined.
• and are either equal or both undefined.
• and are either equal or both undefined.
• and are either equal or both undefined
• and are equal if both are defined.
• If and if both and are defined, then .
• If and and if both and are defined, then .

In general, all laws of arithmetic are valid in as long as all occurring expressions are defined.

Miscellaneous

Several functions can be continuously extended to by taking limits. For instance, one defines etc.

Some discontinuities may additionally be removed. For example, the function can be made continuous (under some definitions of continuity) by setting the value to for , and for and . The function can not be made continuous because the function approaches as approaches 0 from below, and as approaches from above.

Compare the projectively extended real line, which does not distinguish between and . As a result, on one hand a function may have limit on the projectively extended real line, while in the affinely extended real number system only the absolute value of the function has a limit, e.g. in the case of the function at . On the other hand

and

correspond on the projectively extended real line to only a limit from the right and one from the left, respectively, with the full limit only existing when the two are equal. Thus and cannot be made continuous at on the projectively extended real line.