as shown in the following. Continuity of a function for metric spaces and at every point of an interval (i.e., continuity of on the interval ) is expressed by a formula starting with quantifications , (metrics and are and for for the set of real numbers ). For uniform continuity, the order of the first, second, and third quantifications (, , and ) are rotated: . Thus for continuity on the interval, one takes an arbitrary point of the interval, and then there must exist a distance , while for uniform continuity, a single must work uniformly for all points of the interval, Properties Every uniformly continuous function is continuous, but the converse does not hold. Consider for instance the continuous function where is the set of real numbers. Given a positive real number , uniform continuity requires the existence of a positive real number such that for all with , we have . But and as goes to be a higher and higher value, needs to be lower and lower to satisfy for positive real numbers and the given . This means that there is no specifiable (no matter how small it is) positive real number to satisfy the condition for to be uniformly continuous so is not uniformly continuous. Any absolutely continuous function (over a compact interval) is uniformly continuous. On the other hand, the Cantor function is uniformly continuous but not absolutely continuous. 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 this 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 . Examples and nonexamples Examples Linear functions are