Definition:Bounded Mapping/Real-Valued

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This page is about Bounded Mapping in the context of Real-Valued Function. For other uses, see Bounded.

Definition

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


Definition 1

$f$ is bounded on $S$ if and only if:

$f$ is bounded above on $S$

and also:

$f$ is bounded below on $S$.


Definition 2

$f$ is bounded on $S$ if and only if:

$\exists K \in \R_{\ge 0}: \forall x \in S: \size {\map f x} \le K$

where $\size {\map f x}$ denotes the absolute value of $\map f x$.


Definition 3

$f$ is bounded on $S$ if and only if:

$\exists a, b \in \R_{\ge 0}: \forall x \in S: \map f x \in \closedint a b$

where $\closedint a b$ denotes the (closed) real interval from $a$ to $b$.


Function Attaining its Bounds

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

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

Suppose that:

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


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


Unbounded

$f$ is unbounded if it is not bounded.


Also see


Sources