1.4Continuous Functions

Let $f$ be a complex-valued function defined on $\Omega \subset \mathbb{C}$.

1. $f$ is continuous at $z_0 \in \Omega$ if for every $\varepsilon > 0$, there exists $\delta > 0$ such that:

   $$|f(z) - f(z_0)| < \varepsilon \quad \text{whenever } z \in \Omega \text{ and } |z - z_0| < \delta$$

2. Equivalently, $f$ is continuous at $z_0$ if for every sequence $z_n \to z_0$ in $\Omega$, we have $f(z_n) \to f(z_0)$.

A continuous function $f$ on a compact set $\Omega \subset \mathbb{C}$ is bounded and attains both a maximum and a minimum value on $\Omega$ (where maximum/minimum are defined in terms of the real-valued modulus $|f(z)|$).