Thread: show that h is differentiable everywhere and that h' is continuous everywhere but....

Define h(x)=x^3sin(1/x) for x not equal to 0 and h(0)=0. Show that h is differentiable everywhere and that h' is continuous everywhere but fails to have a derivative at one point.

I have h'(x) to be : (3 x^2) sin(1/x)-x cos(1/x) but im i don't know how to show what is being asked.

To be continuous everywhere, at the possible point of discontinuity, if the left hand limit and right hand limit are equal to the function at that point, then the function is continuous everywhere.

To be differentiable everywhere, at the possible point of no-differentiability, the left and right hand limits of the derivative at that point have to be equal.