real analysis - Is a differentiable function required to have at least one point where its derivative is continuous?

Let $f$ there be a real-valued differentiable function everywhere in the interval $]a,b[$.

Does $\frac{df}{dx}$ need to be continuous somewhere in the interval $]a,b[$? Or can a differentiable function $f$ exist so that $\frac{df}{dx}$ is continuous nowhere in the interval $]a,b[$?

Answer

$$f'(x) = \lim_{n\to\infty} n ( f(x+1/n) - f(x) )$$

This is the limit of continuous functions and thus Baire class $1$, that is, it is the pointwise limit of continuous functions.

A theorem states that a pointwise limit of continuous functions can only have discontinuities at a meagre set and thus must be continuous on a dense set of points.