How to show $g(t,u)= \frac{f(t,tu)}{t}$ is $C^\infty$?

Question

Could you give me a hint:
Let $f:\mathbb{R}^2\rightarrow \mathbb{R}$ be a $C^\infty$ function with $f(0,0)=0.$ Define $g(t,u)= f(t,tu)/t$ for $t\neq 0$ and $0$ when $t=0.$ How I will show that $g$ is also $C^\infty$ for $(t,u)\in \mathbb{R}^2.$

Answer 1

A hint: Consider the auxiliary function $$\phi(\tau):=f(\tau\, t,\tau\, t u)\qquad(0\leq\tau\leq1)\ $$ and bring $\phi'(\tau)$ into the game.