# Definition:Bounded Below Mapping/Real-Valued

This page is about Bounded Below in the context of Real-Valued Function. For other uses, see Bounded Below.

## Definition

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:

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

That is, if and only if the set $\set {\map f x: x \in S}$ is bounded below in $\R$ by $L$.

## Unbounded Below

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

Then $f$ is unbounded below on $S$ if and only if it is not bounded below on $S$:

$\neg \exists L \in \R: \forall x \in S: L \le \map f x$

## Examples

### $\tan x$ on $x \in \hointr 0 {\dfrac \pi 2}$

The real tangent function on the half-open interval $\hointr 0 {\dfrac \pi 2}$:

$\forall x \in \hointr 0 {\dfrac \pi 2}: \tan x$

is bounded below by $0$, but unbounded above.