# Definition:Bounded Below Mapping

*This page is about mappings which are bounded below. For other uses, see Definition:Bounded Below.*

## Definition

Let $f: S \to T$ be a mapping whose codomain is an ordered set $\left({T, \preceq}\right)$.

Then $f$ is said to be **bounded below (in $T \ $)** by the lower bound $L$ if and only if:

- $\forall x \in S: L \preceq f \left({x}\right)$

That is, iff $f \left({S}\right) = \left\{{f \left({x}\right): x \in S}\right\}$ is bounded below by $L$.

### Real-Valued Function

The concept is usually encountered where $\left({T, \preceq}\right)$ is the set of real numbers under the usual ordering $\left({\R, \le}\right)$:

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

Then $f$ is **bounded below on $S$** by the lower bound $L$ if and only if:

- $\forall x \in S: L \le \map f x$

## Unbounded Below

Let $f: S \to T$ be a mapping whose codomain is an ordered set $\left({T, \preceq}\right)$.

Then $f$ is **unbounded below (in $T \ $)** if and only if there exists no $L \in S$ such that:

- $\forall x \in S: L \preceq f \left({x}\right)$