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
- 1975: W.A. Sutherland: Introduction to Metric and Topological Spaces ... (previous) ... (next): $5$: Compact spaces: $5.5$: Continuous maps on compact spaces
- 1977: K.G. Binmore: Mathematical Analysis: A Straightforward Approach ... (previous) ... (next): $\S 7.13$