Given a domain $U\subset\mathbb{C}$ $a\in U$ and $f:U\setminus\{a\}\to\mathbb{C}$ 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 $\mathbb{C}$ Put another way, a holomorphic function can come in an arbitrarily small neighborhood of its essential singularity arbitrarily close to any complex value.