Uniform continuity

From Exampleproblems

In mathematical analysis, a function f(x) is called uniformly continuous if, roughly speaking, small changes in the input x affect small changes in the output f(x) ("continuity"), and furthermore the size of the changes in f(x) depends only on the size of the changes in x but not on x itself ("uniformity"). In other words, the function never has an infinite slope.

Continuity itself is a local property of a function—that is, a function f is continuous, or not, at a particular point, and 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 a function. A function is uniformly continuous, or not, on an entire interval, and may be continuous at each point of an interval without being uniformly continuous on the entire interval.

Contents 1 Definition 2 Properties 3 Generalization to topological vector spaces 4 Generalization to uniform spaces

Definition

Given a metric spaces X with distance function , if then a function is called uniformly continuous if for every real number ε > 0 there exists δ > 0 such that for all with d(x,y) < δ, we have that d(f(x),f(y)) < ε.

If X is the domain of real numbers, can be substituted for the standard Euclidian norm, , yeilding the definition: if for all ε > 0 there exists a δ > 0 such that | x − y | < δ implies | f(x) − f(y) | < ε.

Properties

Every uniformly continuous function is continuous, but the converse is not true. Consider for instance the function f(x) = 1/x with domain the positive real numbers. This function is continuous, but not uniformly continuous, since as x approaches 0, the changes in f(x) grow beyond any bound.

If M is a compact metric space, then every continuous f : M → N is uniformly continuous (this is the Heine-Cantor theorem). In particular, if a function is continuous at every point of a closed bounded interval, it is uniformly continuous on that interval.

Every Lipschitz continuous map between two metric spaces is uniformly continuous.

If (x_n) is a Cauchy sequence and f is a uniformly continuous function, then (f(x_n)) is also a Cauchy sequence.

Generalization to topological vector spaces

In the special case of two topological vector spaces V and W, the notion of uniform continuity of a map becomes : for any neighborhood B of zero in W, there exists a neighborhood A of zero in W such that implies .

Generalization to uniform spaces

The most natural and general setting for the study of uniform continuity are the uniform spaces. A function f : X → Y between uniform space 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 and that continuous maps on compact uniform spaces are automatically uniformly continuous.