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

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.

Also see

• Definition:Lower Bound of Real-Valued Function

• Definition:Bounded Above Real-Valued Function
• Definition:Upper Bound of Real-Valued Function

• Definition:Bounded Real-Valued Function

• Results about bounded below real-valued functions can be found here.

Sources

• 1947: James M. Hyslop: Infinite Series (3rd ed.) ... (previous) ... (next): Chapter $\text I$: Functions and Limits: $\S 3$: Bounds of a Function
• 1977: K.G. Binmore: Mathematical Analysis: A Straightforward Approach ... (previous) ... (next): $\S 7.13$
• 1998: David Nelson: The Penguin Dictionary of Mathematics (2nd ed.) ... (previous) ... (next): bound: 1. (of a function)
• 2008: David Nelson: The Penguin Dictionary of Mathematics (4th ed.) ... (previous) ... (next): bound: 1. (of a function)