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Math Help - uniform continuity

  1. #1
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    uniform continuity

    Let f be continuous on [a,b). Prove that lim_{x \rightarrow b^-}f(x) exists iff f is uniformly continuous on [a,b).
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  2. #2
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    If we know \displaystyle\lim_{x\rightarrow b-}f(x) exists and we want to show f is actually uniformly continuous, it helps to invoke a nice theorem.

    Theorem: If f is a continuous real function on a closed and bounded set of real numbers, then f is in fact uniformly continuous.

    Then if we can define a value of f at b which maintains continuity, we've got it!


    Spoiler:
    Use the sequential definition of continuity, i.e. f is continuous if and only if x_n \rightarrow x implies f(x_n)\rightarrow f(x)
    .

    Conversely if the function is uniformly continuous then a straightforward use of the definition shows the limit exists.

    Spoiler:
    Let \epsilon>0 be given and let x_n be any sequence converging to x. Then there exists \delta>0 such that |x-y|<\delta implies |f(x)-f(y)|<\epsilon. Use the Cauchy criterion.
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