Definition:Supremum of Mapping/Real-Valued Function

Definition
Let $f$ be a real-valued function defined on a non-empty subset of the real numbers $S \subseteq \R$.

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

Then 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 $f$

Note that this supremum always exists by the Continuum Property.

Linguistic Note
The plural of supremum is suprema, although the (incorrect) form supremums can occasionally be found if you look hard enough.

Also see

 * Definition:Infimum of Mapping