Set

f(x) = {x sin(1/x), x not = 0

f(x) = {0, x = 0

and g(x) = xf(x)

Assume we know that f is continuous at 0 but not differentiable there, and that g is differentiable at 0. Both f and g are differentiable at each x not = 0.

(a) Find f'(x) and g'(x) for x not = 0.

(b) Show that g' is not continuous at 0.

I can find the f'(x) does not exist exist at 0 and g'(x) equals to 0, but I do not know how to prove something when says is not at a point, help me please.