Definition:Bounded Mapping/Real-Valued/Unbounded

Definition

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

Example: $\paren {-1^n} n$

The function $f$ defined on the integers $\Z$:

$\forall n \in \Z: \map f n := \paren {-1}^n n$

is unbounded.

Example: $\frac 1 x$

The function $f$ defined on the positive real numbers $\openint 0 \to$:

$\forall x \in \openint 0 \to: \map f x := \dfrac 1 x$

is bounded below (by $0$) but unbounded above.

Hence $f$ is unbounded.

Example: $x \sin x$

The function $f$ defined on the positive real numbers $\openint 0 \to$:

$\forall x \in \openint 0 \to: \map f x := x \sin x$

is both unbounded below and unbounded above.

Hence $f$ is unbounded.

Also see

• Equivalence of Definitions of Unbounded Real-Valued Function

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