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I am asking a generalization of the following problem. Given a measure zero set $E$, we can find $f$ so that $F(x)=\int^x_0 f(t) dt$ has $F'(x)=\infty$ on $E$. In particular, by outer regularity, let $V_n \supseteq E$ be open and $|V_n|<2^{-n}$. Then $f(x)=\sum_{n=1} \chi_{V_n}$ does the trick.

But this means that $F'(x)=\infty$ on a little more than $E$, possibly, since we approximated $E$ by a Borel set. Is it possible to find $f$ such that $F'(x)=\infty$ precisely on $E$, given any $E$ with measure zero? Can we characterize the set $\{x:F'(x)=\infty\}$?

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No, it's not possible. Philosophically speaking, you never get anything non-Borel from a continuous function such as your $F$. But I don't have time to write a detailed proof right now. You may want to read the 2nd reference in the article on Zahorski's theorem, and a more recent paper by Fowler and Preiss. –  user31373 Aug 7 '12 at 20:15

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