Which functions of one variable are derivatives ? 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.
 A: 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. 
A: 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.]
A: Take a look a this book by Andrew M. Bruckner: Differentiation of real functions.
Chapter seven is about The problem of characterizing derivatives.
There is a review by Daniel Waterman.
You might also want to take a look at Homeomorphisms in Analysis by Goffman, Nishiura and Waterman.
A: A result that is related to your question (the "almost everywhere" is the difference) :
Every Henstock-Kurzweil integrable function on [a,b] is almost everywhere the derivative of a differentiable function, and inversely, any derivative is Henstock-Kurzweil integrable.
More here : http://www.math.vanderbilt.edu/~schectex/ccc/gauge/
