Definition:Continuous Complex Function/Epsilon-Neighborhood

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 $a \in A_1$.

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.

Also see

• Equivalence of Definitions of Continuous Complex Function

• Results about continuous complex functions can be found here.

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