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This problem is taken from Problem 2.4.31 (page 84) from Problems in Mathematical Analysis: Integration by W. J. Kaczor, Wiesława J. Kaczor and Maria T. Nowak.


Give an example of a bounded function $f:[0,1] \to \mathbb{R}$ which is not Riemann Integrable, but is a derivative of some function $g$ on $[0,1]$.

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    Have you seen [Volterra's function](http://en.wikipedia.org/wiki/Volterra%27s_function)?2010-08-12
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    @Akhil Matthew: Yes i did have a look. But out of ideas.2010-08-12
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    @Chandru: what's lacking? Volterra's function has exactly the properties you request. @Akhil: the link is wrong.2010-08-12
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    @Akhil, @Nate: I fixed the link.2010-08-12

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I gave an answer to this question on Math Overflow some months ago:

https://mathoverflow.net/questions/6711/integrability-of-derivatives/6716#6716

See, in particular, the following paper:

http://math.uga.edu/~pete/Goffman77.pdf