# Definition:Bounded Mapping/Real-Valued

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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

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

## 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.