Removable singularities

Theorem 10.1 (Removable singularities). Let \(f\) have an isolated singularity at \(z_0\). The following are equivalent:

(i)

\(z_0\) is a removable singularity;

(ii)

\(\lim _{z \to z_0} f(z)\) exists and is finite;

(iii)

\(f\) is bounded in some punctured disc \(0 < \abs {z - z_0} < \delta \).

In that case, setting \(f(z_0) = \lim _{z \to z_0} f(z)\) makes \(f\) analytic at \(z_0\).

Source: [1, ch. 5]