Is a holomorphic function with nonvanishing derivative almost injective?

Question

Let $\Omega \subseteq \mathbb C$, be an open, bounded, connected, contractible subset with smooth boundary.

Let $f:\Omega \to \mathbb C$ be holomorphic, and suppose that its derivative $f'$ is everywhere non-vanishing.

Is true that for almost every $y \in f(\Omega)$, $f^{-1}(y)$ is a singleton?

I guess that one could build a counter example by using $g \times g$, where $g: \mathbb S^1 \to \mathbb S^1$ is given by $g(\theta)=2\theta$, but I am not sure.

Answer 1

A counterexample can be found by letting $f(z) = z^2$ and $f$ is defined on $\Omega$ which

• contains $\pm 1$,
• does not contain $0$, and
• is open, bounded, connected, contractible subset with smooth boundary.

There are open neighborhoods $U_\pm$ of $\pm 1$ in $\Omega$ such that $$ f|_{U_\pm} :U_\pm \to f(U_\pm)$$ is a biholomorphism.