The fragment below is from the book 'Theory of functions of a complex variable', by Carathéodory.
The situation is this. We have an analytic function $f(z)$ in some region $G_z\subset\mathbb{C}$. Then $G_w=f(G_z)$ is a region in the $w$-plane. We assume that $G_w^*$ is a simply-connected subregion of $G_w$ that does not contain points $w=f(z)$ such that $f'(z)=0$. We know,
by the inverse function theorem, that $f$ has a local inverse around such points. But then
Carathéodory claims (highlighted below) that we can start at any such point $w_0$ and continue analytically such a local inverse along any "polygonal train" — (i.e any polygonal path) starting from $w_0=f(z_0)$ (as long as we stay inside $G_w^*$). He then deduces — by the monodromy theorem — that $f$ has a single-valued analytic inverse defined on the whole of $G_w^*$.
Carathédory's argument says this:
If $f$ is an analytic function defined on a region $\Omega$ such that $f'(z)\neq 0$ for every $z\in\Omega$, then $f$ has a single-valued analytic inverse defined in any simply connected subregion of $f(\Omega)$.
As far as I know, there are examples of functions defined in simply connected regions that cannot be analytically continued.
So my question is this:
— Is the statement above true? How does Carathédory know that we can extend a local inverse analytically along every polygonal path that lies in $G_w^*$?
Added
In order to clarify the question, here is the final comment Carathéodory
makes:
Best Answer
This is not true without additional hypotesis. If it were true, it would imply Caratheodory's claim about the existence of a single valued inverse, which is false in general:
Let $f(z)=z^2(z-1)$ on $\mathbb{C}$. This map is surjective, and it has only two singular points: $0$ and $\frac23$. However, if we remove this values from the domain, we get a surjective map from $\mathbb{C}-\{0,\frac23\}$ to $\mathbb{C}$ which is non injective. Using the fact that $\mathbb{C}-\{0,\frac23\}$ has $\mathbb{D}$ as its universal cover, we get a map $\varphi:\mathbb{D}\twoheadrightarrow \mathbb{C}$ such that $\varphi'\neq 0$, $\varphi$ is surjective and not injective: thus $\varphi$ does not admit a single valued inverse on $G_w^*=Gw=\mathbb{C}$ as such a function should be entire and bounded and thus by Liouville's theorem, constant.
One can also construct $\varphi:\mathbb{C}\twoheadrightarrow \mathbb{C}$ such that $\varphi'\neq 0$, $\varphi$ is surjective and non linear (and thus non injective, as the only injective entire functions are linear): see this mathoverflow post for such an example (the error function is an explicit example).
$\varphi$ does not admit a single valued inverse on $G_w^*=G'w=\mathbb{C}$, as such a function should be entire and it should avoid more than one point (since for almost every $w, \varphi^{-1}(w)$ is infinite) and thus by Picard's little theorem it should be constant.
If, however, we let $f$ satisfy a few more assumptions, the result follows. For example, if $f:G_z\to G_w^*$ is proper, it is a well known result that $f$ is actually a covering map. As $G_w^*$ is simply connected, it's its own universal cover, and thus the holomorphic lifting lemma implies the existence of an inverse and thus in particular the possibility of analytic continuation along polygonal paths.