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Math Help - Uniformly continuous function

  1. #1
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    Uniformly continuous function

    Let
    f : R--->R be differentiable with bounded derivative. Show that f is uniformly continuous.

    (f(x) - f(y)) / (x - y) = f'(p)
    for some p in (x,y) (by the mean value theorem) and if M > 0 is the bound
    on all values of the derivative, so that
    |f(x) - f(y) | / |x - y| <= M
    or |f(x) - f(y)| <= M*|x-y| for all x,y.


    Can you help please? Thanks

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  2. #2
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    Quote Originally Posted by charikaar View Post
    Let
    f : R--->R be differentiable with bounded derivative. Show that f is uniformly continuous.

    (f(x) - f(y)) / (x - y) = f'(p)
    for some p in (x,y) (by the mean value theorem) and if M > 0 is the bound
    on all values of the derivative, so that
    |f(x) - f(y) | / |x - y| <= M
    or |f(x) - f(y)| <= M*|x-y| for all x,y.


    Can you help please? Thanks

    You're on the right track...

    Fix e>0, we want to show there exists a d such that:

    lx-yl < d implies lf(x) - f(y)l < e

    you already have

    lf(x) - f(y)l <= lx-yl *M

    if lx-yl < d, then

    lf(x) - f(y)l < dM

    Finding the right d is not hard to get lf(x) - f(y)l < e.
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  3. #3
    Math Engineering Student
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    Quote Originally Posted by charikaar View Post
    (f(x) - f(y)) / (x - y) = f'(p)
    for some p in (x,y) (by the mean value theorem) and if M > 0 is the bound
    on all values of the derivative, so that
    |f(x) - f(y) | / |x - y| <= M
    or |f(x) - f(y)| <= M*|x-y| for all x,y.


    Can you help please? Thanks

    [/LEFT]
    the last line of your work implies that f is Lipschitz, and every Lipschitz function is uniformly continuous.
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