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

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## Definition

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

Let $T$ be a subset of $S$.

Suppose that:

$\exists a, b \in T: \forall x \in S: f \left({a}\right) \le f \left({x}\right) \le f \left({b}\right)$

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