2.17 Bounded Functions

If $E$ is a subset of the domain of $f$ , we say $f$ is bounded on $E$ (by $M$ ) if there exists a positive number $M$ such that for all $x$ in $E$
$| f (x) | < M$ Otherwise, $f$ is said to be unbounded.

Geometrically the above definition means that $f$ is bounded on $E$ if the graph of $f$ that is above all $x$ in $E$ lies between some horizontal lines $y = M$ and $y = - M$ .

For example, $f (x) = ⌊ x ⌋$ is bounded on the interval $E = (- 1, 1)$ because for all $x$ in $E$ , $f (x)$ is either 0 or $- 1$ ; that is,
$| f (x) | \leq 1$ but $f (x) = ⌊ x ⌋$ is unbounded on the entire set of real number $R = (- \infty, \infty)$ .