rather strong condition. It is desirable to have a weaker condition from which to deduce extendability. For example, suppose is a real number. At the precalculus level, the function can be given a precise definition only for rational values of (assuming the existence of qth roots of positive real numbers, an application of the Intermediate Value Theorem). One would like to extend to a function defined on all of . The identity shows that is not uniformly continuous on the set of all rational numbers; however for any bounded interval the restriction of to is uniformly continuous, hence Cauchy-continuous, hence extends to a continuous function on . But since this holds for every , there is then a unique extension of to a continuous function on all of . More generally, a continuous function whose restriction to every bounded subset of is uniformly continuous is extendable to , and the converse holds if is locally compact. A typical application of the extendability of a uniformly continuous function is the proof of the inverse Fourier transformation formula. We first prove that the formula is true for test functions, there are densely many of them. We then extend the inverse map to the whole space using the fact that linear map is continuous; thus, uniformly continuous. Generalization to topological vector spaces In the special case of two topological vector spaces and , the notion of uniform continuity of a map becomes: for any neighborhood of zero in , there exists a neighborhood of zero in such that implies For linear transformations , uniform continuity is equivalent to continuity. This fact is frequently used implicitly in functional analysis to extend a linear map off a dense subspace of a Banach space. Generalization to uniform spaces Just as the most natural and general setting for continuity is topological spaces, the most natural and general setting for the study of uniform continuity are the uniform spaces. A function between uniform spaces is called uniformly continuous if for every entourage in there exists an entourage in such that for every in we have in . In this setting, it is also true that uniformly continuous maps transform Cauchy sequences into Cauchy sequences. Each compact Hausdorff space possesses exactly one uniform structure compatible with the topology. A consequence is a generalization of the Heine-Cantor theorem: each continuous function from a compact Hausdorff space to a uniform space