Monodromy theorem

This article gives the statement, and possibly proof, of a basic fact in complex analysis.
View a complete list of basic facts in complex analysis

Statement

Suppose $U\subset \mathbb{C}$ is an open subset and ${\gamma }_1,{\gamma }_2$ are two smooth curves from $p$ to $q$ that are smoothly homotopic. Suppose $V\ni p$ is an open subset contained inside $U$ and $f:V\to \mathbb{C}$ is a holomorphic function.

Let ${\gamma }_t,t\in [0,1]$ denote the intermediate curves for the homotopy between ${\gamma }_0$ and ${\gamma }_1$. Then, suppose the following is true:

For every $t$, we can extend $f$ to a holomorphic function $f(t)$ on an open subset $V_t$ containing ${\gamma }_t$

Then any two ${\gamma }_t$s agree on the overlap and thus, all the values ${\gamma }_t(q)$ are equal.