Global maximum and global minimum

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

• The point $a$ is a global maximum (or absolute maximum) of $f$ on $E$ if

  $$\forall x\in E,\ f(x)\le f(a).$$

• The point $a$ is a global minimum (or absolute minimum) of $f$ on $E$ if

  $$\forall x\in E,\ f(a)\le f(x).$$

Global extrema are stronger than local extrema and need not exist in general. A central theorem in analysis is that continuous functions on compact sets attain both a global maximum and a global minimum.

Examples:

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