TITLE: Which functions of one variable are derivatives ?
QUESTION [44 upvotes]: 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.

REPLY [15 votes]: Here are a few characterizations of derivatives:

D. Preiss and M. Tartaglia
On Characterizing Derivatives
Proceedings of the American Mathematical Society, Vol. 123, No. 8 (Aug., 1995), 2417-2420.
Chris Freiling, On the problem of characterizing derivatives, 
Real Analysis Exchange 23 (1997/98), no. 2, 805-812. 
Brian S. Thomson, On Riemann Sums
Real Analysis Exchange 37 (2011/12), 1-22. [You can download the PDF file here.]

The problem was first posed by W. H. Young.  We include in our article about the Youngs a full quote stating his problem;
Bruckner, Andrew M. and  Thomson, Brian S. 
Real variable contributions of G. C. Young and W. H. Young.
Expo. Math. 19 (2001), no. 4, 337–358. [You can download the PDF file here.]