Past exam of the mathematics course of the University of Cambridge / 2026 / ii / Paper 1 / 24G / c

Since $V$ is simply connected, $g$ has a primitive $G_0$. On connected $U\cap V$, $G_0-F$ has zero derivative and is constant. Adjust that constant to obtain $G=F$ on the overlap, hence the required continuation.

Differentiate a germ of a continuation of $F$ to obtain a germ of the continuation of $f$. This defines a surjective cover $R_{\tilde{F}}\to R_F$. Two primitive germs above the same derivative germ differ by an additive period. Translation by periods gives all deck transformations and acts transitively on each fibre, so the cover is regular.