Why is the absolute sign needed in the definition of a bounded function A function $f$ is bounded if there exists a real number $M$ such that $|f(x)| \le M$ for all $x \in \operatorname{dom}(f)$. Why is the absolute sign needed?

Here is why: 
Suppose we let $f(x) = \begin{cases} x & x < 0 \\ 0 & x \geq 0 \end{cases}$.
Should such a function be considered "bounded"?  It is certainly bounded from above (by $1$ for example), so $f(x) \leq 1$ is true for all $x$...but it isn't bounded from below.  $\lim \limits_{x \to - \infty} f(x) = - \infty$, which means our function is not bounded from below.
This is remedied by saying $|f(x)| \leq M$, because then by definition, $|f(x)| \leq M$ implies $-M \leq f(x) \leq M$, so that $f$ is bounded from above and below.
