# Differentiable Greatest Integer Function

## 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

## 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
Science Advisor
Homework Helper

## 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

## 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?

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
Science Advisor
Homework Helper
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?

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

Dick
Science Advisor
Homework Helper
If it's equal to an integer then it would not be differentiable.

Why not? You have to give reasons.

Because it's discontinuous at all integers.

Dick
Science Advisor
Homework Helper
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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