# Definition:Bounded Mapping/Real-Valued/Unbounded

## Definition

Let $S$ be a set.

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

$f$ is unbounded if and only if it is neither bounded above nor bounded below.

## Examples

### Example: $-1^n n$

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

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

is unbounded.