\begin{definition}[Definition:Bounded Mapping/Real-Valued/Attaining its Bounds]
Let $f: S \to \R$ be a bounded real-valued function.
Let $T$ be a subset of $S$.
Suppose that:
:$\exists a, b \in T: \forall x \in S: \map f a \le \map f x \le \map f b$
Then $f$ '''attains its bounds on $T$'''.
\end{definition}