Can we say anything about the set of all points of discontinuities of $f^{(1)}$,the derivative of a differentiable function on $\mathbb R$.

Question

We all know that for a differentiable function $f$ on $\mathbb R$,its derivative function can have only essential or $2$nd kind discontinuity.Now my question is if $X$ be that set of all points of discontinuities of the derivative function,then can we predict the nature of $X$,i.e.can we say anything about the structure of $X\subset \mathbb R$.