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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 $(t,u)\in \mathbb{R}^2.$

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    Hint: We only need to check whether the derivatives $\partial_t^k g$ exist and are continuous on $\{t=0\}$. You can use induction and just compute.2012-04-25

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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.