Branch point theorem

Statement

Suppose $U \subset C$ is an open subset, $z_{0} \in U$ is a point and $f : U \to C$ is a holomorphic function. Suppose $n$ is the order of zero of the function $z \mapsto f (z) - f (z_{0})$ at $z_{0}$ : in other words, the smallest positive $n$ such that $f^{(n)} (z_{0}) \neq 0$ . Assume $n$ is finite, i.e. $f$ is not constant around $z_{0}$ .

Then, there exist radii $r_{1}, r_{2} > 0$ such that:

For any $w$ such that $| w - f (z_{0}) | < r_{2}$ , the set:

${z \in U ∣ | z - z_{0} | < r_{1}, f (z) = w}$

has cardinality exactly $n$ .

Related facts

Open mapping theorem, which can be viewed as a corollary of the branch point theorem.