Riemann’s removable singularity theorem

Let $U\subset ℂ$ 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

$$\underset{z\to a}{lim}(z-a)f(z)=0.$$

In particular, $a$ is a removable singularity of $f$ if $f$ is http://planetmath.org/node/Boundedbounded near $a$, i.e. if there is a punctured neighborhood $V$ of $a$ and a real number $M>0$ such that for all $z\in V$.