Uniform Continuity - Properties

Properties

Every uniformly continuous function is continuous, but the converse does not hold. Consider for instance the function . Given an arbitrarily small positive real number, uniform continuity requires the existence of a positive number such that for all with, we have . But

and for all sufficiently large x this quantity is greater than .

The image of a totally bounded subset under a uniformly continuous function is totally bounded. However, the image of a bounded subset of an arbitrary metric space under a uniformly continuous function need not be bounded: as a counterexample, consider the identity function from the integers endowed with the discrete metric to the integers endowed with the usual Euclidean metric.

The Heine–Cantor theorem asserts that every continuous function on a compact set is uniformly continuous. In particular, if a function is continuous on a closed bounded interval of the real line, it is uniformly continuous on that interval. The Darboux integrability of continuous functions follows almost immediately from the uniform continuity theorem.

If a real-valued function is continuous on and exists (and is finite), then is uniformly continuous. In particular, every element of, the space of continuous functions on that vanish at infinity, is uniformly continuous. This is a generalization of the Heine-Cantor theorem mentioned above, since .