Definition:Supremum of Mapping/Real-Valued Function

Definition
Let $f$ be a mapping defined on a subset of the real numbers $S \subseteq \R$.

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

It follows from the Continuum Property that the codomain of $f$ has a supremum on $S$.

Thus:
 * $\displaystyle \sup_{x \in S} f \left({x}\right) = \sup f \left({S}\right)$

Also see

 * Infimum