A bounded isolated singularity is removable; boundedness, continuous extension, holomorphic extension, and $(z-z_0)f(z)\to0$ are equivalent.Suggest a slogan
Let \(\Omega\subseteq\CC\) be open, \(z_0\in\Omega\), and let \(f\) be holomorphic on \(\Omega\sm\ts{z_0}\). If \(f\) is bounded on some punctured neighborhood of \(z_0\), then \(z_0\) is a removable singularity of \(f\).
More generally, the following are equivalent:
\(f\) extends holomorphically over \(z_0\): there is a holomorphic \(F\colon\Omega\to\CC\) with \(\ro{F}{\Omega\sm\ts{z_0}}=f\);
\(f\) extends continuously over \(z_0\);
\(f\) is bounded on some punctured neighborhood of \(z_0\);
\(\lim_{z\to z_0}(z-z_0)f(z)=0\).