Simply connected corollary of the monodromy theorem

The <monodromy theorem> gives equality of continuations that exist under its hypothesis. It does not assert that every <holomorphic function> admits <analytic continuation> everywhere, even within a <simply connected domain>.

For an explicit obstruction, take $f(z)=1/(1-z)$ near $a=0$ and attempt to apply the theorem on $D=\mathbb C$. The <path> $\eta(t)=t$ ends at $1$, where no terminal <germ of a holomorphic function> exists. If a <holomorphic> representative $G$ near $1$ continued $f$, then the <identity theorem> would extend $(1-z)G(z)=1$ to that <neighbourhood>. Substituting $z=1$ would give $0=1$. Thus the <pole> prevents the existence hypothesis, although $\mathbb C$ is a <simply connected domain>. The <analytic continuation> article explains why the same <rational function> does continue throughout $\mathbb C\setminus\{1\}$.

Existence at an endpoint by some route is also weaker than existence along every <path> from the base point. Along the real segment from $1$ towards $0$, a <holomorphic square root> of $z$ tends to $0$, yet cannot continue holomorphically through $0$: a local representative $G$ would satisfy $G^2=z$ by the <identity theorem>, and differentiating at $0$ would give $2G(0)G'(0)=1$ with $G(0)=0$. A finite limiting value therefore does not remove a <branch point>. More severe obstructions appear at a <natural boundary of a holomorphic function>.


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 Incoming links: Existence hypothesis of the monodromy theorem