Let $D \subset \mathbb{C}$ be a domain and let $a \in D$. Suppose $f: D \smallsetminus \{a\} \to \mathbb{C}$ is analytic and that $a$ is an essential singularity of $f$. Show that $f$ cannot be univalent (= injective) in any neighborhood of $a$.
This is a trivial consequence of the Picard theorem. But I don't know if there is any elementary approach.