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Math Help - Analysis Question

  1. #1
    Super Member Aryth's Avatar
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    Analysis Question

    Suppose  f : (a,b) \to R is differentiable and | f'(x)|\leq M, \forall x \in (a,b). Prove that f is uniformly continuous on (a,b).

    I am having trouble finding a starting point. Any help would be appreciated.
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  2. #2
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    Re: Analysis Question

    Quote Originally Posted by Aryth View Post
    Suppose  f : (a,b) \to R is differentiable and | f'(x)|\leq M, \forall x \in (a,b). Prove that f is uniformly continuous on (a,b).
    Use the mean value theorem.
    If (x,y)\subseteq (a,b) then \left( {\exists z \in (x,y)} \right)\left[ {\left| {\frac{{f(y) - f(x)}}{{y - x}}} \right| = \left| {f'(z)} \right|} \right]
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