\begin{definition}[Definition:Bounded Mapping/Real-Valued/Definition 3]
Let $f: S \to \R$ be a real-valued function.
$f$ is '''bounded on $S$''' {{iff}}:
:$\exists a, b \in \R_{\ge 0}: \forall x \in S: \map f x \in \closedint a b$
where $\closedint a b$ denotes the (closed) real interval from $a$ to $b$.
\end{definition}