How to prove this statement in calculus

Question

$f(x)$ derivative in $x=0$,and $f(0)=0$, I need to prove that there is a $g(x)$, that is Continuous in $x=0$, and $f(x)=xg(x)$.

what I did is $f'(x)=g(x)+xg'(x)\implies f'(0)=g(0)+0$, but I dont think that the solution is too simple like this.

Answer 1

Simply define$$g(x)=\begin{cases}\frac{f(x)}x&\text{ if }x\neq0\\f'(0)&\text{ otherwise.}\end{cases}$$and prove that $g$ is continuous at $0$.