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Differentiable and uniformly continuous?

  1. Nov 10, 2008 #1
    differentiable and uniformly continuous??

    1. The problem statement, all variables and given/known data
    Suppose f:(a,b) -> R is differentiable and | f'(x) | <= M for all x in (a,b). Prove f is uniformly continuous on (a,b).

    2. Relevant equations
    The definition of uniform continuity is:
    for any e there is a d s.t. | x- Y | < d then | f(x) -f(y | < e.

    3. The attempt at a solution
    Intuitively, if f is differentiable it is continuous. If its derivative is bounded it cannot change fast enough to break continuity. The interval is bounded, and the function must be bounded on the open interval. It seems that there is not way that the function cannot be uniformly continuous. But how do I say that? Or am I on the wrong track altogether.
  2. jcsd
  3. Nov 10, 2008 #2
    Re: differentiable and uniformly continuous??

    Watch this:
    |f(x) -f(y)| = \frac{|f(x) -f(y)|}{|x -y|} \cdot |x-y| < M \cdot \delta

    Can you verify these steps?
  4. Nov 10, 2008 #3
    Re: differentiable and uniformly continuous??

    So long | x-y | not equal 0.

    Yeah, I got the same thing. Actually a little bit differently, I used the mean value theorem, because you don't know that the function is bounded, only its derivative. The mean value theorem gets you to the point where you know that f '(c) = [f(x) - f(y)] / (x-y) is <= M (because f '(x) <= M for all x). Then you just multiply both sides by | x - y | and get the same end result. As long as | x - y| < e/M |f(x) - f(y) | < e.

    Thank you for your help!
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