Definition:Bounded Above Mapping/Real-Valued

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

$f$ is bounded above on $S$ by the upper bound $H$ iff:
 * $\forall x \in S: f \left({x}\right) \le H$

That is, iff the set $\left\{{f \left({x}\right): x \in S}\right\}$ is bounded above in $\R$ by $H$.

Also see

 * Definition:Upper Bound of Real-Valued Function


 * Definition:Bounded Below Real-Valued Function
 * Definition:Lower Bound of Real-Valued Function


 * Definition:Bounded Real-Valued Function