Global maximizer and global minimizer

Let $f:E\to\mathbb{R}$ and let $a\in E$ .

The point $a$ is a global maximizer (or absolute maximizer) of $f$ on $E$ if $\forall x\in E,\ f(x)\le f(a).$ The value $f(a)$ is then the global maximum (or absolute maximum) of $f$ on $E$ .

The point $a$ is a global minimizer (or absolute minimizer) of $f$ on $E$ if $\forall x\in E,\ f(a)\le f(x).$ The value $f(a)$ is then the global minimum (or absolute minimum) of $f$ on $E$ .

Remarks

Global extrema are stronger than local extrema and need not exist in general. The extreme value theorem states that continuous real-valued functions on compact spaces attain both a global maximum and a global minimum.

Examples

On $E=[0,1]$ , $f(x)=x$ has global minimizer $0$ , global maximizer $1$ , minimum value $0$ , and maximum value $1$ .
On $E=(0,1)$ , $f(x)=x$ has no global maximizer or minimizer and no maximum or minimum value.
On $E=\mathbb{R}$ , $f(x)=x^2$ has global minimizer $0$ and minimum value $0$ , but no global maximizer or maximum value.