# Definition:Continuous Complex Function

*This page is about Continuous Mapping in the context of Complex Analysis. For other uses, see Continuous Mapping.*

## Definition

As the complex plane is a metric space, the same definition of continuity applies to complex functions as to metric spaces.

Let $A_1, A_2 \subseteq \C$ be subsets of the complex plane.

Let $f: A_1 \to A_2$ be a complex function from $A_1$ to $A_2$.

Let $a \in A_1$.

### Definition using Limit

$f$ **is continuous at (the point) $a$** if and only if:

- The limit of $\map f z$ as $z \to a$ exists, and
- $\ds \lim_{z \mathop \to a} \map f z = \map f a$

### Epsilon-Delta Definition

$f$ is **continuous at (the point) $a$** if and only if:

- $\forall \epsilon > 0: \exists \delta > 0: \forall z \in A_1: \cmod {z - a} < \delta \implies \cmod {\map f z - \map f a} < \epsilon$

### Epsilon-Neighborhood Definition

Let $A_1$ be open in $\C$.

$f$ is **continuous at (the point) $a$** if and only if:

- $\forall \map {\NN_\epsilon} {\map f a}: \exists \map {\NN_\delta} a: f \sqbrk {\map {\NN_\delta} a} \subseteq \map {\NN_\epsilon} {\map f a}$

where $\map {\NN_\epsilon} a$ is the $\epsilon$-neighborhood of $a$ in $A_1$.

That is, for every $\epsilon$-neighborhood of $\map f a$ in $\C$, there exists a $\delta$-neighborhood of $a$ in $\C$ whose image is a subset of that $\epsilon$-neighborhood.

### Open Sets Definition

Let $A_1$ be open in $\C$.

$f$ is **continuous** if and only if: