2.4 Continuous Functions

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Continuity of functions of a complex variable is much like continuity of functions of a real variable. In this section we define what it means for a function $f:D\to \mathbb{C}$ on a domain to be continuous.

Definition (Continuity of a Function)

Fix a domain $D\subset \mathbb{C}$. A function $f:D\to \mathbb{C}$ is continuous at a point $c\in D$ if $$\underset{z\to c}{lim}f(z)=f(c)$$ holds.

The limit above is formally defined as follows.

Definition (Limits)

Fix a domain $D\subset \mathbb{C}$ and $f:D\to \mathbb{C}$. Fix also $c\in D$. We say that $$\underset{z\to c}{lim}f(z)=\ell$$ or that $f(z)$ tends to $\ell$ as $z$ tends to $c$, or that the limit of $f(z)$ exists as $z\to c$ if, for all $ϵ>0$, there exists $\delta >0$ such that if $z\in D$ and $0<|z-c|<\delta$ then $|f(z)-\ell |<ϵ$.

That is, $f(z)\to \ell$ as $z\to c$ means that if $z$ is very close and not equal to $c$ then $f(z)$ is very close to $\ell$. Note that in this definition we do not need to know the value of $f(c)$.

Example

Let $f:\mathbb{C}\to \mathbb{C}$ be defined by $$f(z)=\{\begin{array}{ll}1& z\ne 0\\ 0& z=0\end{array}$$ so that $$\underset{z\to 0}{lim}f(z)=1$$ but ${\displaystyle \underset{z\to 0}{lim}f(z)\ne f(0)}$. Thus $f$ is not continuous at 0.

Continuous functions of a complex variable obey the same rules as continuous functions of a real variable. Therefore we can use the same strategies for testing and evaluating limits in complex analysis as we used in real analysis.

Lemma (Limit Laws)

Suppose that ${\displaystyle \underset{z\to c}{lim}f(z)=\ell }$ and ${\displaystyle \underset{z\to c}{lim}g(z)=m}$.

1. ${\displaystyle \underset{z\to c}{lim}f(z)+g(z)=\ell +m}$
2. ${\displaystyle \underset{z\to c}{lim}af(z)=a\ell }$ for any $a\in \mathbb{C}$
3. ${\displaystyle \underset{z\to c}{lim}f(z)g(z)=\ell m}$
4. ${\displaystyle \underset{z\to c}{lim}f(z)/g(z)=\ell /m}$ if $m\ne 0$

Lemma (Polynomial Limits)

If $$f(z)=a_n{z}^n+a_{n-1}{z}^{n-1}+\cdots +a_{1}z+a_0$$ then $f$ is continuous at every $c\in \mathbb{C}$.

Lemma (Rational Limits)

If $f,g$ are polynomials and $g(c)\ne 0$ then $$\underset{z\to c}{lim}{\displaystyle \frac{f(z)}{g(z)}}={\displaystyle \frac{f(c)}{g(c)}}$$