# How do I determine whether a function is bounded?

Oct 27, 2014

A function $f$ is bounded in a subset $U$ of its domain if there exist constants $M , m \in \mathbb{R}$ such that

$m \le f \left(x\right) \le M ,$ for all $x \in U .$

For example,

1. $f \left(x\right) = \sin \left(x\right)$ is bounded in $\mathbb{R}$ because

$- 1 \le \sin \left(x\right) \le 1 ,$ for all $x \in \mathbb{R}$.
2. $f \left(x\right) = {x}^{2}$ is bounded in $\left[0 , 1\right]$ because

$0 \le {x}^{2} \le 1 ,$ for all $x \in \left[0 , 1\right] .$