For any $n \geq 1$, does there exist a real function which has $n$ continuous derivatives everywhere but which has $n+1$ derivatives nowhere? Does there exist a real function which has $n$ derivatives everywhere but where the $n$th derivative is continuous almost nowhere?
I assume this question has been asked already but was unable to find it myself. If it has already been asked, I will remove my question. My guess for the first part of my question is that if you take the $n$th antiderivative of the Weierstrass function, the resulting set of functions will all possess $n$ continuous derivatives everywhere, but will possess $n+1$ derivatives nowhere.