Uniform continuity

In mathematical analysis, a function f(x) is called uniformly continuous if, roughly speaking, small changes in the input x effect 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").

Definition

The formal definition is as follows: a function f : M -> N between metric spaces is called uniformly continuous if for every real number ε > 0 there exists a number δ > 0 such that for all x₁, x₂ in M with d(x₁, x₂) < δ, we have d(f(x₁), f(x₂)) < 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.

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 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.