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Does bounded derivative always imply uniform continuity?

  1. Nov 12, 2011 #1
    I'm working on a problem for my analysis class. Here it is:

    Let f be differentiable on an open subset S of R. Suppose there exists M > 0 such that for all x in S, |f'(x)| ≤ M, i.e. the derivative is bounded. Show that f is uniformly continuous on S.

    I'm not too sure that this question is correct though, as I think I have a counterexample. Let S be the union of (-1,0) and (0, 1), then clearly S is open. Define f(x) = |x| / x for x in S.
    Then if x < 0, f(x) = -1, and if x > 0, f(x) = 1.
    f'(x) = 0 at every x in S, since 0 is not in S, so f is differentiable on S and the derivative is bounded.

    And now f is not uniformly continuous on S, since if we set ε= 1, let δ be arbitrary, and pick x,y close to 0 such that x<0, y>0, and |x - y| < δ, it does not follow that |f(x) - f(y)| < ε. So no δ will work for this ε.

    I'd really appreciate any feedback on my reasoning. Thanks for your time!
     
  2. jcsd
  3. Nov 13, 2011 #2

    micromass

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    This indeed looks like a valid counterexample!!
     
  4. Nov 13, 2011 #3

    Deveno

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    indeed, the problem should read:

    Let f be differentiable on an open interval S of R. Suppose there exists M > 0 such that for all x in S, |f'(x)| ≤ M, i.e. the derivative is bounded. Show that f is uniformly continuous on S.

    so there's something special about intervals and continuity. intervals are connected.
     
  5. Nov 13, 2011 #4
    Thanks for the replies!
     
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