Prove that a fuction is continous and differentiable everywhere, but not at f'=0

[charmedbeauty]

Homework Statement

Prove that the function f:ℝ→ℝ, given by

f(x)={x²sin(1/x) if x≠0, 0 if x=0}

is continuous and differentiable everywhere, but that f' is not continuous at 0.

Homework Equations

The Attempt at a Solution

I thought if a function was differentiable everywhere then it was continuous everywhere?

If not what should I do??

 

[I like Serena]

Hi charmedbeauty!

I guess this is exactly the counter example that shows that differentiability (even everywhere) does not imply continuity of the derivative.

What you would need to do is to check the definitions of continuity and differentiability.
Do you have those handy?

I'll help you along.
A function f is continuous at point c iff $\lim\limits_{x \to c} f(x)=f(c)$.
Does that hold for c=0 in your case?

 

[Dick]

Differentiability DOES imply continuity. f IS continuous. It's f' that is discontinuous.

 

[jd12345]

f' exists for all values of x which implies that f is differentiable and hence f is continuous too
f' not being continuous is a different matter

 

[andrien]

cos(1/x) is not continuous at x=0