# Definition:Bounded Above Mapping

This page is about Bounded Above in the context of Mapping. For other uses, see Bounded Above.

## Definition

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

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

$\forall x \in S: \map f x \preceq H$

That is, if and only if $f \sqbrk S = \set {\map f x: x \in S}$ is bounded above by $H$.

### Real-Valued Function

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

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:

$\forall x \in S: \map f x \le H$

## Unbounded Above

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

Then $f$ is unbounded above on $S$ if and only if it is not bounded above on $S$:

$\neg \exists H \in T: \forall x \in S: \map f x \preceq H$