Let $f: \mathbb{R}^{2} \rightarrow \mathbb{R}$ of class $C^{\infty}$ with $f(x,0)=f(0,y)=0$ for all $x, y \in \mathbb{R}$. prove that there exists $g: \mathbb{R^2} \rightarrow \mathbb{R}$ of class $C^{\infty}$ such that $f(x,y)=xyg(x,y)$ for any $(x,y) \in \mathbb{R}^{2}$.
Since $f$ is of class $C^{\infty}$ then all its derivatives are continuous. I don't know how to prove the existence of the function $g$