Definition:Continuous Complex Function

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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 $a \in A_1$ be a point 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
$\displaystyle \lim_{z \mathop \to a} \map f z = \map f a$


Epsilon-Delta Definition

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

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


Epsilon-Neighborhood Definition

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

$\forall \map {N_\epsilon} {\map f a}: \exists \map {N_\delta} a: \map f {\map {N_\delta} a} \subseteq \map {N_\epsilon} {\map f a}$

where $\map {N_\epsilon} a$ is the $\epsilon$-neighborhood of $a$ in $M_1$.


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 Set Definition

$f$ is continuous if and only if:

for every set $U \subseteq \C$ which is open in $\C$, $\map {f^{-1} } U$ is open in $\C$.