# 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:

$\ds \sup_{x \mathop \in S} \map f x := \sup f \sqbrk S$

where

$\sup f \sqbrk S$ is the supremum in $\R$ of the image of $S$ under $f$.