a)

Suppose f(x)=xg(x) for some function g which is continuous at 0. Prove that f is differentiable at 0 and find f'(0) in terms of g.

b)

Suppose f is differentiable at 0 and that f(0)=0, prove that f(x)=xg(x) for some function g which is continuous at 0. Hint: what happens if you try to write g(x)=f(x)/x ?