Table of contents |
2 Properties 3 Generalization to uniform spaces |
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.
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 (xn) is a Cauchy sequence and f is a uniformly continuous function, then (f(xn)) is also a Cauchy sequence.
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.Definition
Properties
Generalization to uniform spaces