# Definition:Bounded Above Mapping

## Definition

Let $f: S \to T$ be a mapping whose codomain is an ordered set $\left({T, \preceq}\right)$.

Then $f$ is bounded above on $S$ by the upper bound $H$ if and only if:

$\forall x \in S: f \left({x}\right) \preceq H$

That is, if and only if $f \left({S}\right) = \left\{{f \left({x}\right): x \in S}\right\}$ is bounded above by $H$.

### Real-Valued Function

The concept is usually encountered where $\left({T, \preceq}\right)$ is the set of real numbers under the usual ordering $\left({\R, \le}\right)$:

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

$f$ is bounded above on $S$ by the upper bound $H$ if and only if:

$\forall x \in S: f \left({x}\right) \le H$

## Unbounded Above

Let $f: S \to T$ be a mapping whose codomain is an ordered set $\left({T, \preceq}\right)$.

Then $f$ is unbounded above on $S$ iff it is not bounded above on $S$:

$\neg \exists H \in T: \forall x \in S: f \left({x}\right) \preceq H$