Definition:Continuity/Complex Function

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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) $a$ iff:

  • The limit of $f \left({z}\right)$ as $z \to a$ exists
  • $\displaystyle \lim_{z \to a} f \left({z}\right) = f \left({a}\right)$


Epsilon-Delta Definition

$f$ is continuous at (the point) $a$ iff:

$\forall \epsilon > 0: \exists \delta > 0: \left|{z - a}\right| < \delta \implies \left|{f \left({z}\right) - f \left({a}\right)}\right| < \epsilon$


Epsilon-Neighborhood Definition

$f$ is continuous at (the point) $a$ iff:

$\forall N_\epsilon \left({f \left({a}\right)}\right): \exists N_\delta \left({a}\right): f \left({ N_\delta \left({a}\right)}\right) \subseteq N_\epsilon \left({f \left({a}\right)}\right)$

where $N_\epsilon \left({a}\right)$ is the $\epsilon$-neighborhood of $a$ in $M_1$.


That is, for every $\epsilon$-neighborhood of $f \left({a}\right)$ 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 iff:

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