\begin{definition}[Definition:Bounded Mapping/Complex-Valued]
Let $f: S \to \C$ be a complex-valued function.
Then $f$ is '''bounded''' {{iff}} the real-valued function $\cmod f: S \to \R$ is bounded, where $\cmod f$ is the modulus of $f$.
That is, $f$ is '''bounded''' if there is a constant $K \ge 0$ such that $\cmod {f \paren z} \le K$ for all $z \in S$.
\end{definition}