Definition:Supremum of Mapping/Real-Valued Function/Definition 1

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

Let $f$ be bounded above on $S$.

The supremum of $f$ on $S$ is defined by:
 * $\displaystyle \sup_{x \mathop \in S} f \left({x}\right) := \sup f \left[{S}\right]$

where
 * $\sup f \left[{S}\right]$ is the supremum in $\R$ of the image of $S$ under $f$.

Also see

 * Equivalence of Definitions of Supremum of Real-Valued Function