Definition:Bounded Above Mapping

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

That is, $f \sqbrk S = \set {\map f x: x \in S}$ 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 $\struct {\R, \le}$:

Also see

 * Definition:Upper Bound of Mapping


 * Definition:Bounded Below Mapping
 * Definition:Lower Bound of Mapping


 * Definition:Bounded Mapping