Definition: Continuous Functions at Single Complex Numbers

Let \(D\) be a subset of complex numbers \(\mathbb C\), \(a\in D\), and let \(f:D\to\mathbb R\) be a function. As a special case of continuous functions in metric spaces we call \(f\) continuous at the complex number \(a\in\mathbb C\), if and only if

\[\lim_{x\to a} f(x)=f(a).\]

This means that there for every convergent complex sequence \((\xi_n)_{n\in\mathbb N}\), \(\xi\in D\) i.e. a sequence with \(\lim \xi_n=a\), we also have \(\lim f(\xi_n)=f(a)\).

Equivalent Definition

Continuous functions can also be defined using the \(\epsilon\)-\(\delta\) definition of continuity \(f\) is continuous at the point \(a\), if and only if for every \(\epsilon > 0\) there is a \(\delta > 0\) such that \[|f(x)-f(a)| < \epsilon\] for all \(x\in D\) with \[|x-a| < \delta.\]

In the above definition, the general distance for metric spaces has been replaced by the absolute value of complex numbers.

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  1. Forster Otto: "Analysis 1, Differential- und Integralrechnung einer Veränderlichen", Vieweg Studium, 1983