Continuity of Complex Functions

Continuity of Complex Functions

Recall from the Limits of Complex Functions page that if $A \subseteq \mathbb{C}$, $z_0$ is an accumulation point of $\mathbb{C}$, and $f : A \to \mathbb{C}$ then the limit of $f$ as $z$ approaches $z_0$ equals $A$ denoted $\displaystyle{\lim_{z \to z_0} f(z) = A}$ if for all $\epsilon > 0$ there exists a $\delta > 0$ such that if $z \in A$ and $\mid z - z_0 \mid < \delta$ then $\mid f(z) - A \mid < \epsilon$.

With the definition of a limit of a complex function defined we can now also define the concept of continuity of a function $f$ at a point $z_0$.

Definition: Let $A \subseteq \mathbb{C}$ and let $f : A \to \mathbb{C}$. Then $f$ is said to be Continuous at $z_0 \in A$ if $\displaystyle{\lim_{z \to z_0} f(z) = f(z_0)}$. If $f$ is continuous for every point $z_0 \in A$ then $f$ is said to be Continuous on $A$. If $f$ is not continuous at $z_0$ then $f$ is said to be Discontinuous at $z_0$.

Many of the basic functions that we come across will be continuous functions. For example, all polynomials are continuous on $\mathbb{C}$, i.e., $f(z) = a_0 +a_1z^1 + a_2z^2 + ... + a_nz^n$ is continuous at every $z_0 \in \mathbb{C}$.

For a more complicated example, consider the following function:

(1)
\begin{align} \quad f(z) = \frac{z^2 + 2}{1 + z^2} \end{align}

This is a rational function. Notice that the numerator of this function is simply a polynomial and is continuous at every $z_0 \in \mathbb{C}$. Problems arise when the denominator equals $0$ though for which $f$ becomes undefined. Notice that $1 + z^2 = 0$ if and only if $z = \pm i$. So $f$ is actually discontinuous at $z = \pm i$ since $f(\pm i)$ is not even defined!

We will now state some of the familiar properties regarding continuous functions.

Theorem 1: Let $f$ and $g$ be continuous functions at $z_0 \in \mathbb{C}$. Then:
a) $f + g$ is continuous at $z_0$.
b) $f - g$ is continuous at $z_0$.
c) $fg$ is continuous at $z_0$.
d) $\displaystyle{\frac{f}{g}}$ is continuous at $z_0$ provided that $g(z_0) \neq 0$

The proofs of (a) - (d) are omitted but follow from the limit laws for complex functions.

Theorem 2: Let $f : A \to \mathbb{C}$. The following statements are equivalent:
a) $f$ is continuous on $A$.
b) The inverse image of every open set in $\mathbb{C}$ is open in $A$.
c) The inverse image of every closed set in $\mathbb{C}$ is open in $A$.
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