This is motivated by this recent MO question.

Is there a complete characterization of those functions $f:(a,b)\rightarrow\mathbb R$ that are pointwise derivative of some everywhere differentiable function $g:(a,b)\rightarrow\mathbb R$ ?

Of course, continuity is a sufficient condition. Integrability is not, because the integral defines an absolutely continuous function, which needs not be differentiable everywhere. A. Denjoy designed a procedure of reconstruction of $g$, where he used transfinite induction. But I don't know whether he assumed that $f$ is a derivative, or if he had the answer to the above question.