Limits and continuity.

Definition 3.1.1   Let $f$ be a function of the complex variable $z$ . The complex number $l$ is called the limit of $f$ at $z_0$ if $\forall \epsilon >0, \exists \delta >0 s.t. \vert z-z_0\vert < \delta \Rightarrow \vert f(z)-l\vert < \epsilon$ .

We denote:

$$\underset{z \rightarrow z_0}{\lim} f(z)=l.$$

Example 3.1.2   Let $f(z)= 3z-1$ . We prove that $\underset{z \rightarrow 1+i}{\lim} f(z)= 2+3i$ .

Let $\varepsilon >0$ be given. We look for $\delta >0$ such that $\vert f(z)-(2+3i)\vert< \varepsilon$ .

$$\begin{vmatrix}f(z)-(2+3i) \end{vmatrix} < \varepsilon \Longleftrightarrow \begin{vmatrix}3z-1-2-3i \end{vmatrix} < \varepsilon$$

$$\quad \begin{vmatrix}3z-3(1+i) \end{vmatrix} < \varepsilon$$

$$\quad 3 \begin{vmatrix}z-(1+i) \end{vmatrix} < \varepsilon$$

$$\quad \begin{vmatrix}z-(1+i) \end{vmatrix} < \frac {\varepsilon}{3}.$$

We take any $\delta$ such that $0< \delta < \varepsilon /3$ and we are done.

Definition 3.1.3   Let $f$ be a function defined on a domain $R$ in $\mathbb{C}$ and let $z_0$ be an interior point of $R$ . The function $f$ is continuous at $z_0$ if $\underset{z \rightarrow z_0}{\text{lim}} f(z)=f(z_0)$ .

Formally this definition is identical to the corresponding definition in Calculus. Thus we get easily the two following propositions:

Proposition 3.1.4   Let $f$ and $g$ be two functions defined on a neighborhood of $z_0$ . We suppose that $f$ and $g$ are continuous at $z_0$ .

(i)

$f+g$ is continuous at $z_0$ .

(ii)

$fg$ is continuous at $z_0$ .

(iii)

If $g(z_0) \neq 0$ , then $1/g$ is continuous at $z_0$ .

(iv)

If $g(z_0) \neq 0$ , then $f/g$ is continuous at $z_0$ .

For a proof, we suggest to the reader to have a look at his/her course in Calculus. The needed adaptation is merely to understand the absolute value here as the absolute value of complex numbers instead of that of real numbers. The same remark applies to Proposition 1.5.

Proposition 3.1.5   Let $f$ be a function defined on a neighborhood of $z_0$ and let $g$ be a function defined on a neighborhood of $f(z_0)$ . If $f$ is continuous at $z_0$ and if $g$ is continuous at $f(z_0)$ , then $gof$ is continuous at $z_0$ .

Recall that a polynomial function is a function of the form $f:z \mapsto \underset{k=0}{\overset{n}{\sum}}a_nz^n$ , where all the $a_i$ are given complex numbers and $a_n \neq 0$ . We apply the two first properties of Proposition 1.4 to prove Corollary 1.6.

Corollary 3.1.6        A polynomial function is continuous on the whole of $\mathbb{C}$ .

Recall now that a rational function is the quotient of two polynomial functions. Applying the last property of Proposition 1.4 to prove Corollary 1.6, we prove Corollary 1.7.

Corollary 3.1.7        A rational function is continuous at every point of its domain of definition.

Example 3.1.8   The function $f$ such that $f(z)= ((1+i)z-2i)/(z^2+1)$ is continuous on $\mathbb{C} - \{ -i, i \}$ .