A map from a metric space to a metric space is said to be uniformly continuous if for every , there exists a such that whenever satisfy .
Note that the here depends on and on but that it is entirely independent of the points and . In this way, uniform continuity is stronger than continuity and so it follows immediately that every uniformly continuous function is continuous.
Examples of uniformly continuous functions include Lipschitz functions and those satisfying the Hölder condition. Note however that not all continuous functions are uniformly continuous with two very basic counterexamples being (for ) and (for . On the other hand, every function which is continuous on a compact domain is necessarily uniformly continuous.