Definition:Continuous Complex Function/Epsilon-Delta

Definition

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 $a \in A_1$.

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

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

Also see

• Equivalence of Definitions of Continuous Complex Function

• Results about continuous complex functions can be found here.

Sources

• 1998: David Nelson: The Penguin Dictionary of Mathematics (2nd ed.) ... (previous) ... (next): continuous function (continuous mapping)
• 2001: Christian Berg: Kompleks funktionsteori $\S \text {i}.1$
• 2008: David Nelson: The Penguin Dictionary of Mathematics (4th ed.) ... (previous) ... (next): continuous function (continuous mapping, continuous map)