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Definition: Continuous Complex Functions
Let $U\subseteq\mathbb C$ be a subset of complex numbers and let $f:U\to\mathbb C$ be a function $f$ is called continuous at $\alpha\in U$ if $f$ converges to $\alpha$ and $$\lim_{z\to\alpha} f(z)=f(\alpha).$$ $f$ is said to be continuous in $U$, if $f$ is continuous at every point $\alpha\in U.$
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References
Bibliography
 Forster Otto: "Analysis 1, Differential und Integralrechnung einer Veränderlichen", Vieweg Studium, 1983