i know that each smooth function is differentiable which follows from the fact that if partial derivatives exist at each point in a domian then f is differentiable everywhere on that domain. but does the converse true i.e. is a differentiable function is necessarily a smooth one ?
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What do you mean by "smooth"? – Giuseppe Negro May 25 '14 at 15:05
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partial derivatives of all orders exist and continuous – user153271 May 25 '14 at 15:07
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1Then it is false. Take the function $f(x)=x\lvert x \rvert$ for $x\in \mathbb{R}$. It is differentiable but has derivatives only up to first order. – Giuseppe Negro May 25 '14 at 15:13
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It is false whatever (reasonable) you mean with smooth: even if for you smooth means just $C^{1}$ there exists differentiable functions whose derivative is not continuous and thus are not $C^{1}$. An example is $f:\mathbb{R}\mapsto\mathbb{R}$ defined by $$f(x)=\left\{\begin{array}{ll}x^2\sin\left(\frac{1}{x}\right)&x\neq0\\0&x=0\end{array}\right.$$ It is differentiable, $f'(0)=0$, but $f'$ is not continuous.
Dario
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