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 $\{x:F'(x)=\infty\}$?