Differentiable Greatest Integer Function

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Homework Statement


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

Homework Equations





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.
 

Answers and Replies

  • #2
Dick
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Homework Statement


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

Homework Equations





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.

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?
 
  • #3
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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?
 
  • #4
Dick
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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?

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?
 
  • #5
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If it's equal to an integer then it would not be differentiable.
 
  • #6
Dick
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If it's equal to an integer then it would not be differentiable.

Why not? You have to give reasons.
 
  • #7
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Because it's discontinuous at all integers.
 
  • #8
Dick
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Because it's discontinuous at all integers.

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?
 
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