Definition:Maximum Value of Real Function/Absolute

Definition
Let $f: S \to \R$ be a real-valued function.

Let $f$ be bounded above by a supremum $B$.

It may or may not be the case that $\exists x \in S: f \left({x}\right) = B$.

If such a value exists, it is called the maximal value or maximum of $f$ on $S$, and that this maximum is attained at $x$.

Also see

 * Definition:Minimal Value