real analysis - Find a differentiable $f$ such that $f'$ is not continuous.

I'm trying to solve this problem:

Find a differentiable function $f:\mathbb{R} \longrightarrow \mathbb{R}$ such that $f':\mathbb{R} \longrightarrow \mathbb{R}$ is not continuous at any point of $\mathbb{R}$.

Any hints would be appreciated.

Answer

You are looking for a derivative that is discontinuous everywhere on $\Bbb R$. Such a function doesn't exist. Since $f'$ is the pointwise limit of continuous functions, it is a Baire class $1$ function. A theorem of Baire says that the set of discontinuities of $f'$ is a meager subset of $\Bbb R$.