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Continuity#

Definition: Continuity of Complex Functions

Let \(f: \mathcal{D} \subseteq \mathbb{C} \to \mathbb{C}\) be a complex function.

A complex function \(f: \mathcal{D} \subseteq \mathbb{C} \to \mathbb{C}\) is continuous at \(c \in \mathcal{D}\) if and only if its limit for \(z \to c\) is equal to its value there.

\[ \lim_{z \to c} f(z) = f(c) \]

We say that \(f\) is continuous on \(S\subseteq \mathcal{D}\) if it is continuous at each \(z \in S\). Moreover, if \(S = \mathcal{D}\), we just say that \(f\) is continuous.

Theorem: Continuity of the Real and Imaginary Parts

Let \(f: \mathcal{D} \subseteq \mathbb{C} \to \mathbb{C}\) be a complex function.

If \(f\) is continuous at \(c \in \mathbb{C}\), then its real part \(\operatorname{Re} f\) and real part \(\operatorname{Re} f\) part \(\operatorname{Im} f\) are also continuous at \(c\).

Proof

TODO

Theorem: Continuity of Sum, Product and Division

Let \(f\) and \(g\) be complex function.

If \(f\) and \(g\) are continuous at \(c \in \mathbb{C}\), then so are

  • \(\lambda f + \mu g\) for all \(\lambda, \mu \in \mathbb{C}\);
  • \(fg\);
  • \(f/g\), provided that \(g(c) \ne 0\).
Proof

TODO

Theorem: Continuity of Composition

Let \(f\) and \(g\) be complex function.

If \(g\) is continuous at \(c \in \mathbb{C}\) and \(f\) is continuous at \(g(c)\), then their composition \(f\circ g\) is also continuous at \(c\).

Proof

TODO

Bibliography#

  1. N. H. Asmar, L. Grafakos, "Analytic Functions," in Complex Analysis with Applications, Columbia, MO, USA: Springer, 2018