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

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

## Definition

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

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

The infimum of $f$ on $S$ is defined as $\displaystyle \inf_{x \mathop \in S} \map f x := k \in \R$ such that:

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