Lusin's theorem for derivatives with respect to a continuous function

For a nowhere constant continuous function $g$ on a real interval $I$ and for a Borel measure $\mu$ on $I$, we give simple necessary and sufficient conditions guaranteeing, for any Borel function $f$ on $I$, the existence of a continuous function $F$ on $I$ such that the derivative of $F$ with respect to $g$ is, $\mu$ almost everywhere, equal to $f$.