Suppose that $f$ is differentiable at 0, and that $f(0) = 0$. Prove that $f(x) = xg(x)$ for some function $g$ which is continuous at 0.
This is a problem from Spivak's Calculus, namely problem 27 of Chapter 10. (This is not homework, but rather self-study.) I am not sure how to go about this proof. The hint given in the text is to consider that $g(x)$ can be written as $f(x)/x$, but this puzzles me, because then continuity of $g$ at 0 says that $\lim_{x \to 0} g(x) = g(0) = f(0)/0 = 0/0$.