\begin{definition}[Definition:Bounded Mapping/Real-Valued/Definition 2]
Let $f: S \to \R$ be a real-valued function.
$f$ is '''bounded on $S$''' {{iff}}:
:$\exists K \in \R_{\ge 0}: \forall x \in S: \size {\map f x} \le K$
where $\size {\map f x}$ denotes the absolute value of $\map f x$.
\end{definition}