is a differentiable function is necessarily a smooth one

Question

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 ?

Answer 1

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.