Riemann’s theorem on isolated singularities

Let the complex function $f(z)$ be holomorphic in a deleted neighbourhood of the point  $z=z_0$  of the closed complex plane$ℂ\cup \{\mathrm{\infty }\}$.  This point is

• a regular point (or a removable singularity) iff $f(z)$ is bounded in a neighbourhood of $z_0$,

• a pole iff  $\underset{z\to z_0}{lim}|f(z)|=+\mathrm{\infty }$,

• an essential singularity iff there is neither of the above cases.