Definition: Limits of Complex Functions

Riesz F., Sz.-Nagy, B.: "Functional Analysis (Translation)", Dover Publications, 1955
Kneser, Hellmuth: "Mathematische Lehrbücher - Funktionentheorie", Vanderhoeck &amp; Ruprecht, 1958
Lang, Serge: "Complex Analysis", Springer, 1999, Forth Edition

Let $U\subseteq\mathbb C$ be a subset of complex numbers and let $f:U\to\mathbb C$ be a function $f(x)$ is said to converge to a limit at the point $\alpha$, formally $$w=\lim_{\substack{z\in U\\z\to\alpha}} f(z)$$

if given $\epsilon > 0$ there exist $\delta > 0$ such that if $$|z-\alpha| < \delta,\quad z\in U,$$ then $$|f(z)-w| < \epsilon.$$

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