Casorati-Weierstrass theorem

Given a domain $U\subset ℂ$, $a\in U$, and $f:U\setminus \{a\}\to ℂ$ being holomorphic, then $a$ is an essential singularity of $f$ if and only if the image of any punctured neighborhood of $a$ under $f$ is dense in $ℂ$. Put another way, a holomorphic function can come in an arbitrarily small neighborhood of its essential singularity arbitrarily close to any complex value.