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Continuity of the derivative of a decreasing differentiable function

  1. Oct 24, 2012 #1
    1. The problem statement, all variables and given/known data

    To solve a problem in a book, I need to know whether or not the following is true:

    Let f be a real-valued, decreasing differentiable function defined on the interval [itex] [1, \infty)[/itex] such that [itex]\lim_{x \rightarrow \infty} f(x) = 0[/itex]. Then the derivative of f is continuous.

    2. Relevant equations

    N/A

    3. The attempt at a solution
    Tried working directly with the definitions (decreasing function, continuity, derivative), but got nowhere. Consulted various references (such as baby Rudin and Hobson's Theory of Functions of a Real Variable), with no luck. Spent time Googling combinations of the various terms, again with no results.

    I suspect that the above statement is true. The canonical example of a differentiable function whose derivative is not continuous (see here) is not monotone in any interval containing the origin. However, I can't find either a proof or a counterexample.
     
  2. jcsd
  3. Oct 24, 2012 #2

    Dick

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    Your canonical example may not be monotone near the origin, but if you change a bit you can make one that is. Suppose you subtract say, 3x?
     
  4. Oct 25, 2012 #3
    Thanks for the suggestion! By subtracting 3x from the function, we subtract 3 from the derivative. That makes the derivative negative and so the original function is monotone decreasing. I'll have to think about how to apply that to my original question.
     
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