TITLE: Spivak's Calculus exercise. Chapter 10, Problem 27
QUESTION [7 upvotes]: 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 $\lim_{x \to 0} g(x) = g(0) = f(0)/0 = 0/0$.

REPLY [7 votes]: What is $\displaystyle \lim_{x \to 0} \frac{f(x) - f(0)}{x-0}$ ?