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Differentiable Greatest Integer Function

  1. Dec 8, 2012 #1
    1. The problem statement, all variables and given/known data
    k(x)=x2*[1/x] for 0<x≤1
    k(x)=0 for x=0
    Find where k(x) is differentiable and find the derivative
    2. Relevant equations



    3. The attempt at a solution
    I know that it is differentiable for all ℝ\Z on (0,1], but I am unsure how to find the derivative for this problem.
     
  2. jcsd
  3. Dec 8, 2012 #2

    Dick

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    If you mean what R\Z usually means then R\Z on (0,1] is (0,1). I suspect you mean something else. Suppose 1/x is between two integers, say n<1/x<n+1?
     
  4. Dec 8, 2012 #3
    Yes sorry that was a typo, should be (0,1). So would I set k=[1/x], which would make f(x)=x2*k

    which would imply that f'(x)=2xk
    Is this what you mean?
     
  5. Dec 8, 2012 #4

    Dick

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    Sure. So if 1/x is between two integers then your function is differentiable, yes? Suppose 1/x is equal to an integer? Then what?
     
  6. Dec 8, 2012 #5
    If it's equal to an integer then it would not be differentiable.
     
  7. Dec 8, 2012 #6

    Dick

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    Why not? You have to give reasons.
     
  8. Dec 8, 2012 #7
    Because it's discontinuous at all integers.
     
  9. Dec 8, 2012 #8

    Dick

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    True if you mean f(x) is discontinuous when 1/x is an integer. You should probably say that in a more proofy way, like saying what the one sided limits are of f(x) or using a theorem. But I think the main point of the exercise is what happens at x=0, since they bothered to define f(0)=0. f(x) might have a one-sided derivative at x=0. Does it?
     
    Last edited: Dec 8, 2012
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