# Definition:Bounded Mapping/Real-Valued

## 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:

$\exists K \in \R_{\ge 0}: \forall x \in S: \left\vert{f \left({x}\right)}\right\vert \le K$

where $\left\vert{f \left({x}\right)}\right\vert$ denotes the absolute value of $f \left({x}\right)$.

### 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.