This question asks whether there exists a function that has a derivative that is discontinuous everywhere. Is there any function that has a derivative almost everywhere, but where the derivative function is everywhere discontinuous, for the definition of continuity that takes into account only the inputs where the derivative is defined?
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Take a bounded measurable function on $[0,1]$ that has no continuity point even ae in the sense that for every $x \in [0,1]$ there is no $g$ continuous at $x$ st $g=f$ ae on a small interval near $x$.
To construct such, one finds $E$ measurable st for for every subinterval $I \subset [0,1]$ both $I \cap E, I \cap E^c$ have non zero measure and take $f=\chi_E$.
Then $F(x)=\int_0^xf(t)dt$ is differentiable ae and $F'=f$ ae so $F'$ restricted to its set of differentiability points is definitely not continuous at any differentiability point since for any $x$ where $F'$ exist and any interval $I$ containing $x$, we have $F'=1$ on a set of positive measure in $I$ and $F'=0$ also on a set of positive measure in $I$, so $\liminf_{y \to x, F' \text {exists at y}}F'(y)=0$ and $\limsup_{y \to x, F' \text {exists at y}}F'(y)=1$
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