Definition:Locally Uniform Convergence
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Definition
Let $f_n : \C \supseteq U \to \C$ be a sequence of functions.
For $u \in U$ let $D_r(u)$ be the disk of radius $r$ about $u$.
If, for each $x \in U$ there is $r > 0$ such that $f_n$ converges uniformly on $D_r(x)$ (and $D_r(x)$ is contained in $U$).
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