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Wikipedia / Uniform continuity

In mathematics, a real function of real numbers is said to be uniformly continuous.

The difference between uniform continuity and (ordinary) continuity is that in uniform continuity there is a globally applicable, while in (ordinary) continuity there is a locally applicable. So uniform continuity is a stronger continuity condition than continuity; a function that is uniformly continuous is continuous but a function that is continuous is not necessarily uniformly continuous. The concepts of uniform continuity and continuity can be expanded to functions defined between metric spaces.

on the real (number) line. However, any Lipschitz map between metric spaces is uniformly continuous, in particular any isometry (distance-preserving map).

Although continuity can be defined for functions between general topological spaces, defining uniform continuity requires more structure. The concept relies on comparing the sizes of neighbourhoodss of distinct points, so it requires a metric space, or more generally a uniform space.

Definition for functions on metric spaces

with metric spaces, the following definitions of uniform continuity and (ordinary) continuity hold.

Definition of uniform continuity

is called uniformly continuous if for every real number by the definition of a neighbourhood in a metric space.

are subsets of the real line can be the standard one-dimensional Euclidean distance is a material conditional").
. Here quantifications) are used.
is uniformly continuous if it admits a modulus of continuity.

Definition of (ordinary) continuity

is called continuous if for every real number.
.
is a monotonically non-decreasing function.

Local continuity versus global uniform continuity

In the definitions, the difference between uniform continuity and continuity. Continuity is a local, and this can be determined by looking at only the values of the function in an arbitrarily small neighbourhood of that point. When we speak of a function being continuous on an interval, we mean that the function is continuous at every point of the interval. In contrast, uniform continuity is a global. On the other hand, it is possible to give a definition that is local), although it is not possible to give a local definition of uniform continuity for an arbitrary hyperreal-valued function, see below.

are structurally similar as shown in the following.

for metric spaces at every point ) is expressed by a formula starting with quantifications

for the set of real numbers).

For uniform continuity, the order of the first, second, and third quantifications) are rotated:

of the interval,,

of the interval,

Properties

Every uniformly continuous function is continuous is the set of real numbers. But

is not uniformly continuous.

Any absolutely continuous function (over a compact interval) is uniformly continuous. On the other hand, the Cantor function is uniformly continuous but not absolutely continuous.

The image of a totally bounded subset under a uniformly continuous function is totally bounded. However, the image of a bounded subset of an arbitrary metric space under a uniformly continuous function need not be bounded: as a counterexample, consider the identity function from the integers endowed with the discrete metric to the integers endowed with the usual Euclidean metric.

The Heine–Cantor theorem asserts that every continuous function on a compact set is uniformly continuous. In particular, if a function is continuous on a (bounded) closed interval of the real line, it is uniformly continuous on that interval. The Darboux integrability of continuous functions follows almost immediately from this theorem.

that vanish at infinity, is uniformly continuous..

Examples and nonexamples

Examples

Linear functions are the simplest examples of uniformly continuous functions.
is a compact set.
If a function is differentiable on an open interval and its derivative is bounded, then the function is uniformly continuous on that interval.
Every Lipschitz continuous map between two metric spaces is uniformly continuous. More generally, every Hölder continuous function is uniformly continuous.
The absolute value function. This shows uniformly continuous functions are not always differentiable.
Despite being nowhere differentiable, the Weierstrass function is uniformly continuous.
Every member of a uniformly equicontinuous set of functions is uniformly continuous.

Nonexamples

Functions that are unbounded on a bounded domain are not uniformly continuous. The tangent function but is not.
.

Visualization

around that point, then the graph lies completely within the height of the rectangle, i.e., the graph do not pass through the top or the bottom side of the rectangle. For functions that are not uniformly continuous, this isn't possible; for these functions, the graph might lie inside the height of the rectangle at some point on the graph but there is a point on the graph where the graph lies above or below the rectangle. (the graph penetrates the top or bottom side of the rectangle.)

History

The first published definition of uniform continuity was by Heine in 1870, and in 1872 he published a proof that a continuous function on an open interval need not be uniformly continuous. The proofs are almost verbatim given by Dirichlet in his lectures on definite integrals in 1854. The definition of uniform continuity appears earlier in the work of Bolzano where he also proved that continuous functions on an open interval do not need to be uniformly continuous. In addition he also states that a continuous function on a closed interval is uniformly continuous, but he does not give a complete proof.

Other characterizations

Non-standard analysis

In non-standard analysis, a real-valued function of a real variable is microcontinuous at a point is infinitesimal whenever is infinitesimal. Thus is continuous on a set . (see non-standard calculus for more details and examples).

Characterization via sequences

For a function between Euclidean spaces, uniform continuity can be defined in terms of how the function behaves on sequences such that

we have

Relations with the extension problem

, a continuous function. A question to answer: ?

If , the answer is given by the Tietze extension theorem. So it is necessary and sufficient to extend : that is, we may assume without loss of generality that is that it is Cauchy-continuous, i.e., the image under is complete (and thus the completion of ), then every continuous function from to a metric space is Cauchy-continuous. Therefore when is complete, if and only if is Cauchy-continuous.

It is easy to see that every uniformly continuous function is Cauchy-continuous and thus extends to is, as seen above, not uniformly continuous, but it is continuous and thus Cauchy continuous. In general, for functions defined on unbounded spaces like , uniform continuity is a rather strong condition. It is desirable to have a weaker condition from which to deduce extendability.

can be given a precise definition only for rational values of (assuming the existence of qth roots of positive real numbers, an application of the Intermediate Value Theorem. The identity

shows that is not uniformly continuous on the set of all rational numbers; however for any bounded interval the restriction of extends to a continuous function on . But since this holds for every , there is then a unique extension of to a continuous function on all of .

whose restriction to every bounded subset of is uniformly continuous is extendable to , and the converse holds if is locally compact.

A typical application of the extendability of a uniformly continuous function is the proof of the inverse Fourier transformation formula. We first prove that the formula is true for test functions, there are densely many of them. We then extend the inverse map to the whole space using the fact that linear map is continuous; thus, uniformly continuous.

Generalization to topological vector spaces

In the special case of two topological vector spaces

For linear transformations, uniform continuity is equivalent to continuity. This fact is frequently used implicitly in functional analysis to extend a linear map off a dense subspace of a Banach space.

Generalization to uniform spaces

Just as the most natural and general setting for continuity is topological spaces, the most natural and general setting for the study of uniform continuity are the uniform spaces between uniform spaces is called uniformly continuous if for every entourage in there exists an entourage in .

In this setting, it is also true that uniformly continuous maps transform Cauchy sequences into Cauchy sequences.

Each compact Hausdorff space possesses exactly one uniform structure compatible with the topology. A consequence is a generalization of the Heine-Cantor theorem: each continuous function from a compact Hausdorff space to a uniform space is uniformly continuous.