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.
Best Answer
I can't claim much knowledge here, but I am given to understand that the class of differentiable functions (or the class of functions which are derivatives of such) is really quite nasty and complicated. This paper by Kechris and Woodin indicates that there is some very serious descriptive set theory involved: that there is a hierarchy of levels of complication indexed by $\omega_1$ (i.e., the set of countable ordinals). This online article by Kechris and Louveau also looks relevant.