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In the above definition, the general distance for metric spaces has been replaced by the absolute value of complex numbers.

Github: ### References

#### Bibliography

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