Prove differentiability and continuity

[hth]

Homework Statement

Determine that, if f(x) =

{xsin(1/x) if x =/= 0
{0 if x = 0

that f'(0) exists and f'(x) is continuous on the reals. (Sorry I can't type the function better, it's piecewise)

Homework Equations

The Attempt at a Solution

For f'(0) existing,

For x ≠ 0,

(f(x) - f(0))/(x - 0) = (x sin(1/x))/x = sin(1/x).

Sin(1/x) doesn't have a limit as x → 0 because the function oscillates between -1 and 1, f is not differentiable at x = 0. Therefore f'(0) doesn't exist.

For f'(x) being continuous.

For any given ɛ > 0, there exists δ > 0 such that |x - y| < δ and |f(x) - f(y)| < ɛ for all x, y on the reals.

For all x $$\in$$ R (R is the reals), ... and this where I'm stuck.

 

[jgens]

If you know that $f'(0)$ doesn't exist, doesn't that prove that $f'(x)$ is not continuous on the reals?

 

[hth]

Yeah, you're right. Haha, I'm sorry. Total brain letdown on that one.