Realvalued function/Set/Maximum and minimum/Definition

Maximum

Let ${}M$ denote a set, and

$f\colon M\longrightarrow \mathbb {R}$

a function. We say that ${}f$ attains in a point ${}x\in M$ its maximum, if

$f(x)\geq f(x'){\text{ holds for all }}x'\in M,$

and that ${}f$ attains in ${}x$ its minimum, if

$f(x)\leq f(x'){\text{ holds for all }}x'\in M.$