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Math Help - prove that f is uniformly continuous..

  1. #1
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    prove that f is uniformly continuous..

    Suppose f: (a,b) -> R is differentiable and |f'(x)|<=M for all x in (a,b). Prove that f is uniformly continuous on (a,b). Give an example of a function f: (0,1)->R that is differentiable and uniformly continuous on (0,1) but such that f' is unbounded.

    I have no clue.
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  2. #2
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    x\sin(\frac1x) should do for your example.
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  3. #3
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    Let x,t\in \left(a,b \right). By the mean value theorem of derivatives:

    f(x)-f(t)=(x-t)f'(\lambda )

    where \lambda \in (x,t). Hence:

    \left| f(x)-f(t)\right|\leq M\left| x-t\right|

    Now for any \epsilon >0, we can put

    \delta =\frac{\epsilon }{M}

    so that whenever \left| x-t\right|<\delta , we will have \left| f(x)-f(t)\right|<\epsilon . This proves that f(x) is uniformly continuous.
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