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.