Corollary (domain and range are real, heineborel is used implicitly): f uniformly continous is bounded on a bounded set E. This is because there is a continous extension onto a compact set containing E, so applying the max value theorem we see that f is bounded on E.
Lemma: Continuity implies uniform continuity on a compact set. (This is a true fact in general metric spaces)
Characterization of Uniform continuity on bounded subsets of the line: f is uniformly continuous on a bounded set E iff f can be continuously extended to the closure of E.
