absolute extremum

The single highest (or lowest) value $f$ takes over its entire domain, not just near one point.

$f$ has an absolute (global) maximum at $(a,b)$ if $f(a,b)\geq f(x,y)$ for every $(x,y)$ in $f$ 's entire domain (an absolute minimum reverses the inequality) — a claim no purely local test alone can settle. Every absolute extremum is a local extremum, but the reverse fails constantly, and on an unbounded or punctured domain an absolute max or min may simply fail to exist at all, however many local ones $f$ has.

Examples

For $f(x,y)=x^2+y^2$ , $(0,0)$ is the absolute minimum over all of $\mathbb{R}^2$ (value $0$ ), but no absolute minimum exists once the origin is removed from the domain — the infimum $0$ is approached but never attained.