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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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I assume you mean $C^\infty$, i.e. the function has all derivatives? (somehow $\mathbb{C}^\infty$ means the Riemann sphere to me) – Daan Michiels Apr 25 '12 at 8:49
@Makuasi, why did you roll back the edit? I was just trying to improve the question's presentation. – Andy Apr 25 '12 at 8:51
What have you tried? What are you having trouble with? Can you deal with the function away from (0,0)? – Alex B. Apr 25 '12 at 8:54
Probably the last $f$ should be $g$ – Giuseppe Apr 25 '12 at 8:54
yes @Daan, Andy:sorry, I did not get. – Un Chien Andalou Apr 25 '12 at 9:00
up vote 2 down vote accepted

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.

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