TITLE: Why is the absolute sign needed in the definition of a bounded function
QUESTION [8 upvotes]: 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?

REPLY [3 votes]: Case 1:
$f$ is a $\mathbb{R} \rightarrow \mathbb{R}$ function:
The definition can have multiple equivalent forms:
$\exists\ m, M \in \mathbb{R} : \forall\ x \in dom(f) : m \leq f(x) \leq M$
$\exists\ M \in \mathbb{R} : \forall\ x \in dom(f) : - M \leq f(x) \leq M$
$\exists\ M \in \mathbb{R} : \forall\ x \in dom(f) : | f(x) | \leq M$
When teaching calculus it is usually done this way:
$f$ is upper bounded $\leftrightarrow \exists\ M \in \mathbb{R} : \forall\ x \in dom(f) : f(x) \leq M$
$f$ is lower bounded $\leftrightarrow \exists\ m \in \mathbb{R} : \forall\ x \in dom(f) : m \leq f(x)$
$f$ is (generally) bounded $\leftrightarrow$ $f$ is upper and lower bounded
Case 2:
$f$ is a $\mathbb{C} \rightarrow \mathbb{C}$ function:
Since complex numbers can not be compared the only option from the three definitions abowe is the third. That is also correct for real and complex and even more abstract functions if the $|.|$ is correctly defined. For example any euclidean space. So it also works for $\mathbb{R}^n \rightarrow \mathbb{R}^m$ functions.
Note that $|.|$ can be defined in multiple ways. For example:
$\sqrt{(Im z) ^ 2 + (Re z) ^ 2}$
$|Im z| + |Re z|$