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Real Analysis: Uniformly continuous

  1. Mar 11, 2006 #1
    Please help me with this one:

    Suppose f is a continuous real valued function on R - real #s and that 0<f(x)<3 for every x in R. Let F(x) = integral from 0 to x of f(t) dt.

    a) Show that F is uniformly continuous on R
    b) Suppose the continuity assumption is left out, but the function f is still assumed to have 0<f(x)<3 for all x and to be integrable on every bounded interval. Must the conclusion of uniform continuity of F on R still be true? Prove or give counterexample.

    Thank You for your help
  2. jcsd
  3. Mar 11, 2006 #2

    matt grime

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    By teh (second?) Fundamental Theorem of Calculus, F, in part a), is a differentiable function with (uniformly) bounded derivative. It is a standard exercise to show this is uniformly continuous, so what do you think you need to do now you've got this (slightly) different way of thinking about it?
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