Let $f\colon\mathbb{R}\to\mathbb{R}$ be a continuous function with weak derivative (i.e. the derivative in the sense of distribution) in $C^1(\mathbb{R})$. Does this condition imply that $f$ is two times continuously differentiable (i.e. $f\in C^2(\mathbb{R}))$?
A sufficient condition for a function to be of class $C^2$ in the weak sense.
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real-analysis
analysis
distribution-theory