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Let $f:\mathbb{R}^2 \rightarrow \mathbb{R}$ and $g: \mathbb{R} \rightarrow \mathbb{R}$ be continously differentiable. Is then $h:\mathbb{R} \rightarrow \mathbb{R}$ with $h(x)=f(x,g(x))$ differentiable? Well untill know I didn't find a counterexample. How can I prove it by the limit definition?

Thesinus
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    Have you seen the prove of multivariable chain rule? – Li Chun Min Jun 18 '17 at 13:26
  • Yes I have, but it seems very complicated. Isn't there a shorter proof? This question only gives one point, I'll do it anyway with the multivariable chain rule . – Thesinus Jun 18 '17 at 14:24
  • I have tried and it seems a direct proof would be largely similar to the proof to the multivariable chain rule…writing the first degree Taylor's formulas, estimating with the operator norm, multiplying and diving the norm of an intermediate vector, and arguing the remainder is sublinear… – Li Chun Min Jun 18 '17 at 14:34

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