In addition he also states that a continuous function on a closed interval is uniformly continuous, but he does not give a complete proof. Other characterizations Non-standard analysis In non-standard analysis, a real-valued function of a real variable is microcontinuous at a point precisely if the difference is infinitesimal whenever is infinitesimal. Thus is continuous on a set in precisely if is microcontinuous at every real point . Uniform continuity can be expressed as the condition that (the natural extension of) is microcontinuous not only at real points in , but at all points in its non-standard counterpart (natural extension) in . Note that there exist hyperreal-valued functions which meet this criterion but are not uniformly continuous, as well as uniformly continuous hyperreal-valued functions which do not meet this criterion, however, such functions cannot be expressed in the form for any real-valued function . (see non-standard calculus for more details and examples). Cauchy continuity For a function between metric spaces, uniform continuity implies Cauchy continuity . More specifically, let be a subset of . If a function is uniformly continuous then for every pair of sequences and such that we have Relations with the extension problem Let be a metric space, a subset of , a complete metric space, and a continuous function. A question to answer: When can be extended to a continuous function on all of ? If is closed in , the answer is given by the Tietze extension theorem. So it is necessary and sufficient to extend to the closure of in : that is, we may assume without loss of generality that is dense in , and this has the further pleasant consequence that if the extension exists, it is unique. A sufficient condition for to extend to a continuous function is that it is Cauchy-continuous, i.e., the image under of a Cauchy sequence remains Cauchy. If is complete (and thus the completion of ), then every continuous function from to a metric space is Cauchy-continuous. Therefore when is complete, extends to a continuous function if and only if is Cauchy-continuous. It is easy to see that every uniformly continuous function is Cauchy-continuous and thus extends to . The converse does not hold, since the function is, as seen above, not uniformly continuous, but it is continuous and thus Cauchy continuous. In general, for functions defined on unbounded spaces like , uniform continuity is a