# Function such that f'(x) exists but limf'(x) as x -> 0 does not

## Homework Statement

Does anyone have an example of a function such that f'(x) exists everywhere, but limf'(x) as x approaches 0 does not?

## The Attempt at a Solution

I was thinking something along the lines of f(x) = (2x^5 + 5x^2)/x^11
Since then f'(x) = (10x^4 + 10x) / 11x^10
and limf'(x) as x approaches 0 does not exist since 11(0)^10 = 0 and you cannot divide by 0. However, I realized that that means that f'(x) does not exist everywhere since limf'(x) doesn't exist as x approaches 0, that implies that f'(0) does not exist.

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Mark44
Mentor

## Homework Statement

Does anyone have an example of a function such that f'(x) exists everywhere, but limf'(x) as x approaches 0 does not?

## The Attempt at a Solution

I was thinking something along the lines of f(x) = (2x^5 + 5x^2)/x^11
Since then f'(x) = (10x^4 + 10x) / 11x^10
I see two problems with this.
1. Your function f is not defined at x = 0, so f' is also undefined at x = 0. This violates the condition that f' is defined everywhere.
2. Your differentiation is incorrect. The derivative of a quotient is NOT the quotient of the derivatives. To differentiate a quotient you normally need to use the quotient rule. For your example, carrying out the division first before differentiating makes it unnecessary to use the quotient rule.

f(x) = (2x5 + 5x2)/x11 = 2x-6 + 5x-9. Now you can differentiate using the sum rule and the power rule.

In any case, this is not a good example, since f' is not defined everywhere.
and limf'(x) as x approaches 0 does not exist since 11(0)^10 = 0 and you cannot divide by 0. However, I realized that that means that f'(x) does not exist everywhere since limf'(x) doesn't exist as x approaches 0, that implies that f'(0) does not exist.

STEMucator
Homework Helper
So you want a function that's continuous everywhere, but not differentiable at 0.

Mark44
Mentor
So you want a function that's continuous everywhere, but not differentiable at 0.

This one came to my mind, as well, but it doesn't work. For this function, f' doesn't exist at x = 0. The condition in the problem is that f'(x) exists everywhere.

jbunniii
Homework Helper
Gold Member
Hint: Try starting with ##\sin(1/x)## and multiplying by something suitable so that ##f'(0)## exists but ##\lim_{x \rightarrow 0} f'(x)## does not exist.

Bacle2

f(x)=-x2/2 , for x<0

f(0)=0

f(x)=x2/2 for x>0

Then f'(x)=|x| , so that f' is defined everywhere, but the limits at 0 don't agree.

jbunniii
Homework Helper
Gold Member

f(x)=-x2/2 , for x<0

f(0)=0

f(x)=x2/2 for x>0

Then f'(x)=|x| , so that f' is defined everywhere, but the limits at 0 don't agree.
But this ##f'## is continuous everywhere, so the limits will agree everywhere.