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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? – Akhil Mathew Aug 12 '10 at 21:52
@Akhil Matthew: Yes i did have a look. But out of ideas. – anonymous Aug 12 '10 at 21:54
@Chandru: what's lacking? Volterra's function has exactly the properties you request. @Akhil: the link is wrong. – Nate Eldredge Aug 12 '10 at 22:04
@Akhil, @Nate: I fixed the link. – Larry Wang Aug 12 '10 at 22:26
up vote 4 down vote accepted

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

See, in particular, the following paper:

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