Holomorphic extension across an isolated point is equivalent to boundedness, continuity, or $(z-a)f(z)\to0$ there.Suggest a slogan
Let \(U\subseteq \CC\) be open, let \(a\in U\), and let \(f\) be holomorphic on \(U\setminus\theset{a}\). The following are equivalent:
\(f\) extends to a holomorphic function on \(U\);
\(f\) extends to a continuous function on \(U\);
there exists a neighborhood \(V\) of \(a\) such that \(f\) is bounded on \(V\setminus\theset{a}\);
\(\displaystyle\lim_{z\to a} (z-a)f(z) = 0\).
What conditions are equivalent to $f$ extending holomorphically over an isolated singularity?