Definition:Bounded Mapping/Real-Valued

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

Definition

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

Definition 1

$f$ is bounded on $S$ if and only if:

$f$ is bounded above on $S$

and also:

$f$ is bounded below on $S$.

Definition 2

$f$ is bounded on $S$ if and only if:

$\exists K \in \R_{\ge 0}: \forall x \in S: \size {\map f x} \le K$

where $\size {\map f x}$ denotes the absolute value of $\map f x$.

Definition 3

$f$ is bounded on $S$ if and only if:

$\exists a, b \in \R_{\ge 0}: \forall x \in S: \map f x \in \closedint a b$

where $\closedint a b$ denotes the (closed) real interval from $a$ to $b$.

Function Attaining its Bounds

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

Let $T$ be a subset of $S$.

Suppose that:

$\exists a, b \in T: \forall x \in S: \map f a \le \map f x \le \map f b$

Then $f$ attains its bounds on $T$.

Unbounded

Let $S$ be a set.

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

Definition $1$

$f$ is unbounded if and only if it is either unbounded above or unbounded below.

Definition $2$

$f$ is unbounded if and only if:

for every positive real number $M$ there exists $x_M \in \R$ such that:

$\size {\map f {x_M} } > M$

Examples

Sine of $\dfrac 1 x$

The function $g$ defined on the real numbers $\R$:

$\forall x \in \R: g := \map \sin {\dfrac 1 x}$

is bounded.

Also see

• Equivalence of Definitions of Bounded Real-Valued Function

• Definition:Bound of Real-Valued Function

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

Sources

• 2014: Christopher Clapham and James Nicholson: The Concise Oxford Dictionary of Mathematics (5th ed.) ... (previous) ... (next): bounded function