Riemann's removable singularity theorem

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 \(V\setminus\theset{a}\);

• \(\lim_{z\to a} (z-a)f(z) = 0\).