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 P: 176 Corollary (domain and range are real, heine-borel 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.