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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.

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    $\begingroup$ Since differentiable functions are continuous, to be of Baire class $1$ (a pointwise limit of continuous functions) is certainly necessary. $\endgroup$
    – Theo Buehler
    Commented May 30, 2011 at 15:28
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    $\begingroup$ See this wiki page for some partial results: en.wikipedia.org/wiki/… $\endgroup$
    – Mark
    Commented May 30, 2011 at 15:30
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    $\begingroup$ Another necessary condition is mapping intervals into intervals $\endgroup$
    – Pietro Majer
    Commented May 30, 2011 at 23:07
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    $\begingroup$ @Pietro. I mentionned this point in my answer to the previous MO question; in the form a derivative satisfies the intermediate value property. $\endgroup$
    – Denis Serre
    Commented May 31, 2011 at 6:40
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    $\begingroup$ There is a theorem (of Maximoff?) stating that any Baire 1 function which satisfies the intermediate value property is the composition of a derivative and a homeomorphism (and the converse is obvious). This does not answer your question but I think it's cute (I'm pretty sure I read this somewhere in Kechris's "Classical Descriptive Set Theory", but I don't have it with me) $\endgroup$
    – Julien Melleray
    Commented May 31, 2011 at 15:34

4 Answers 4

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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.

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    $\begingroup$ Yeah, that's why nobody ever talks about or works with pointwise differentiable functions. We usually learn about derivatives and differentiability and think about them as being more "elementary" than integration and integrability. But from a theoretical and even computational standpoint, the latter is much easier to work with than the former. $\endgroup$
    – Deane Yang
    Commented May 30, 2011 at 18:08
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Here are a few characterizations of derivatives:

  1. D. Preiss and M. Tartaglia On Characterizing Derivatives Proceedings of the American Mathematical Society, Vol. 123, No. 8 (Aug., 1995), 2417-2420.

  2. Chris Freiling, On the problem of characterizing derivatives, Real Analysis Exchange 23 (1997/98), no. 2, 805-812.

  3. 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.]

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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.

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    $\begingroup$ Andy has updated his account of this problem in a survey article for the Real Analysis Exchange: Bruckner, Andrew M. The problem of characterizing derivatives revisited. Real Anal. Exchange 21 (1995/96), no. 1, 112--133. Download from our web site here: classicalrealanalysis.info/documents/Bruckner1995.rae.1341343228.pdf $\endgroup$
    – B S Thomson
    Commented Dec 28, 2012 at 17:58
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    $\begingroup$ Clickable link: Brucker - The problem of characterising derivatives revisited (MSN). $\endgroup$
    – LSpice
    Commented Nov 7, 2017 at 16:04
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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/

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