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 *x*_{1}, *x*_{2} in *M* with d(*x*_{1}, *x*_{2}) < δ, we have d(*f*(*x*_{1}), *f*(*x*_{2})) < 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 (*x*_{n}) is a Cauchy sequence and *f* is a uniformly continuous function, then (*f*(*x*_{n})) 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 (*x*_{1}, *x*_{2}) in *U* we have (*f*(*x*_{1}), *f*(*x*_{2})) 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.