Riemann’s removable singularity theorem

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

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

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 $|f(z)| for all $z\in V$.

Title Riemann’s removable singularity theorem RiemannsRemovableSingularityTheorem 2013-03-22 13:33:00 2013-03-22 13:33:00 pbruin (1001) pbruin (1001) 4 pbruin (1001) Theorem msc 30D30 Pole EssentialSingularity Meromorphic RiemannsTheoremOnIsolatedSingularities