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Function Monotone on Some Interval

  1. Oct 15, 2005 #1
    I need the following proposition in order to prove another theorem, and I can't seen to find it in my textbook. Any hints on how to proceed, or whether it's actually TRUE, would be helpful.

    "If f is defined and continuous on some interval I, then there exists subintervals I'=[x-a,x+b], for some real numbers a and b, at every point x in I such that f is monotone on I'."
     
    Last edited by a moderator: Oct 15, 2005
  2. jcsd
  3. Oct 15, 2005 #2

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    I don't think that's true. Consider the function:

    [tex]f(x)=\left\{\begin{array}{cc}x \mbox{ sin}(\frac{1}{x}),&\mbox{ if } x \neq 0\\0,&\mbox{ if } x=0\end{array}[/tex]

    This is continuous, but is not monotone on any interval containing 0.
     
    Last edited: Oct 15, 2005
  4. Oct 15, 2005 #3
    Ok, I've changed my approach. What about:

    "If f is defined and continuous on an interval I and c is in I such that f(c)>0, then there exists an interval I' in I where c is in I' such that f(x)>0 for every x in I'."
     
    Last edited by a moderator: Oct 15, 2005
  5. Oct 15, 2005 #4

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    That's true. Just use the epsilon delta definition of continuity, taking delta as f(c).
     
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