An isolated singularity is removable exactly when the function stays bounded near it.Suggest a slogan
Let \(\Omega\subseteq\CC\) be open, let \(z_0\in\Omega\), and let \(f\) be holomorphic on \(\Omega\setminus\{z_0\}\). Then \(z_0\) is a removable singularity of \(f\) if and only if \(f\) is bounded on \(\{z : 0<\abs{z-z_0}<\delta\}\) for some \(\delta>0\).