Let $K$ be a compact subset of the domain of the definition of a holomorphic function $f$. And the function $f$ satisfies the following two conditions:
(1) $f$ is injective on $K$;
(2) $f$ has no critical point on $K$. PS: $z$ is a critical point of $f$, if $f'(z)=0$.
Question: Does there exists a neighborhood of $K$ on which $f$ is injective? Prove or disprove!