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:

$\exists a, b \in T: \forall x \in S: \map f a \le \map f x \le \map f b$


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


Sources