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)=\left\lfloor x\right\rfloor $ 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)|\leq1

\]
but $f(x)=\left\lfloor x\right\rfloor $ is unbounded on the entire set of real number $\mathbb{R=}(-\infty,\infty)$.