Definition:Bounded Mapping/Real-Valued/Attaining its Bounds

Definition
Let $f: S \to \R$ be a bounded real-valued function.

Let $\inf \left({f \left({S}\right)}\right) \in f \left({S}\right)$ and $\sup \left({f \left({S}\right)}\right) \in f \left({S}\right)$.

Then $f$ attains its bounds on $S$.