Differentiable Greatest Integer Function

[kolley]

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.

 

[Dick]

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?

 

[kolley]

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?

 

[Dick]

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?

 

[kolley]

If it's equal to an integer then it would not be differentiable.

 

[Dick]

If it's equal to an integer then it would not be differentiable.

Why not? You have to give reasons.

 

[kolley]

Because it's discontinuous at all integers.

 

[Dick]

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?