Uniform continuity

if a function is continuous on a closed bounded interval of the real line, it is uniformly continuous on that interval

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.

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 is uniformly continuous.