At an essential singularity, the image of every punctured neighborhood is dense in $\CC$.Suggest a slogan
Let \(\Omega\subseteq\CC\) be open, let \(z_0\in\Omega\), and let \(f\) be holomorphic on \(\Omega\setminus\theset{z_0}\) with an essential singularity at \(z_0\). Then for every open set \(V\subseteq\Omega\) containing \(z_0\), the image \(f(V\setminus\theset{z_0})\) is dense in \(\CC\).
If \(f\) is a nonconstant entire function, then \(f(\CC)\) is dense in \(\CP^1\).
State the Casorati-Weierstrass theorem.