Riemann's removable singularity theoremRiemannsRemovableSingularityTheorem2003-04-05 04:44:192003-04-05 04:44:19Theorem% this is the default PlanetMath preamble. as your knowledge
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% define commands hereLet $U\subset\mathbb{C}$ 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
$$
\lim_{z\to a}(z-a)f(z)=0.
$$
In particular, $a$ is a removable singularity of $f$ if $f$ is \PMlinkid{bounded}{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 $z\in V$.