# Definition:Supremum of Mapping/Real-Valued Function/Definition 2

This page is about Supremum of Real-Valued Function. For other uses, see Supremum.

## 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 as $\ds \sup_{x \mathop \in S} \map f x := K \in \R$ such that:

$(1): \quad \forall x \in S: \map f x \le K$
$(2): \quad \exists x \in S: \forall \epsilon \in \R_{>0}: \map f x > K - \epsilon$