Let $U\subset\mathbb{C}$ be a domain, $a\in U$, and let $f:U\setminus\{a\}$ be holomorphic. Then $a$ is a removable singularity of $f$ if and only if

In particular, $a$ is a removable singularity of $f$ if $f$ is bounded near $a$, i.e. if there is a punctured neighborhood $V$ of $a$ and a real number $M>0$ such that $|f(z)|<M$ for all $z\in V$.