Uniform continuity

In mathematics, a function f is uniformly continuous if, roughly speaking, it is possible to guarantee that f(x) and f(y) be as close to each other as we please by requiring only that x and y be sufficiently close to each other; unlike ordinary continuity, where the maximum distance between f(x) and f(y) may depend on x and y themselves.

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

Continuity itself is a local property of a function—that is, a function f is continuous, or not, at a particular point, and this can be determined by looking only at 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 only that it is continuous at each point of the interval. In contrast, uniform continuity is a global property of f, in the sense that the standard definition refers to pairs of points rather than individual points. On the other hand, it is possible to give a definition that is local in terms of the natural extension f* (the characteristics of which at nonstandard points are determined by the global properties of f), although it is not possible to give a local definition of uniform continuity for an arbitrary hyperreal-valued function, see below.

The mathematical statements that a function is continuous on an interval I and the definition that a function is uniformly continuous on the same interval are structurally very similar. Continuity of a function for every point x of an interval can thus be expressed by a formula starting with the quantification

whereas for uniform continuity, the order of the first, second, and third quantifiers are rotated:

Thus for continuity at each point, one takes an arbitrary point x, and then there must exist a distance δ,

while for uniform continuity a single δ must work uniformly for all points x (and y):

Any absolutely continuous function 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 closed bounded interval of the real line, it is uniformly continuous on that interval. The Darboux integrability of continuous functions follows almost immediately from this theorem.

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.[1]

In non-standard analysis, a real-valued function f of a real variable is microcontinuous at a point a precisely if the difference f*(a + δ) − f*(a) is infinitesimal whenever δ is infinitesimal. Thus f is continuous on a set A in R precisely if f* is microcontinuous at every real point a ∈ A. Uniform continuity can be expressed as the condition that (the natural extension of) f is microcontinuous not only at real points in A, but at all points in its non-standard counterpart (natural extension) *A in *R. Note that there exist hyperreal-valued functions which meet this criterion but are not uniformly continuous, as well as uniformly continuous hyperreal-valued functions which do not meet this criterion, however, such functions cannot be expressed in the form f* for any real-valued function f. (see non-standard calculus for more details and examples).

For a function between metric spaces, uniform continuity implies Cauchy continuity (Fitzpatrick 2006). More specifically, let A be a subset of Rn. If a function f : A → Rm is uniformly continuous then for every pair of sequences xn and yn such that

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.

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. A function f : X → Y between uniform spaces is called uniformly continuous if for every entourage V in Y there exists an entourage U in X such that for every (x1, x2) in U we have (f(x1), f(x2)) in V.

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 generalisation of the Heine-Cantor theorem: each continuous function from a compact Hausdorff space to a uniform space is uniformly continuous.