A First Course in Complex Analysis · Problem 10

Examples of Functions — Beck, Marchesi, Pixton & Sabalka

Mathematics — Complex Analysis · continuity

Problem

Determine where each of the following functions

f: \mathbb{C} \to \mathbb{C}

is continuous: (a)

f(z) = \begin{cases} 0 & \text{if } z = 0 \text{ or } |z| \text{ is irrational,} \\ \dfrac{1}{q} & \text{if } |z| = \dfrac{p}{q} \in \mathbb{Q} \setminus \{0\} \text{ (written in lowest terms).} \end{cases}

(b)

f(z) = \begin{cases} 0 & \text{if } z = 0, \\ \sin \phi & \text{if } z = r e^{i\phi} \neq 0. \end{cases}