Continuity

Let $D \subseteq \mathbb{C}$, let $f: D \to \mathbb{C}$ be a function. $f$ is continuous at $z_0 \in D$ iff, for every $\epsilon > 0$, there exists $\delta > 0$ such that, for every $z \in D$,

$f$ is continuous on $A \subseteq D$ iff $f$ is continuous at each point of $A$.

Continuity at a Limit Point

Let $f: D \to \mathbb{C}$, $z_0 \in D$.

• If $z_0$ is not an isolated point of $D$, $f$ is continuous at $z_0$ iff $\displaystyle\lim_{z \to z_0} f(z) = f(z_0)$
• If $z_0$ is an isolated point of $D$, $f$ is continuous at $z_0$

Composition of Continuous Functions

A composition of continuous functions is continuous.

Nonvanishing Near a Point

If $f$ is continuous and non-zero at $z_0$, then $f(z) \ne 0$ throughout some neighborhood of $z_0$.

Boundedness on a Closed and Bounded Set

If $f$ is continuous on a closed and bounded set $R \subseteq \mathbb{C}$, there exists a non-negative real number $M$ such that

for all $z \in R$, with equality for at least 1 point in $R$.