closed, bounded region

A domain with no missing boundary points and no infinite extent — the condition that guarantees a continuous function has an absolute max and min.

A closed, bounded region is a domain that includes its own boundary and does not extend infinitely in any direction. Adding this condition restores, unconditionally, the guarantee that a continuous $f$ attains an absolute maximum and minimum somewhere on the region — a guarantee an unbounded or punctured domain can lose entirely. Finding those extrema means checking two disjoint kinds of candidates: every critical point in the region's interior, and every extremum of $f$ restricted to the boundary curve (typically found by parametrizing the boundary and reducing to a single-variable candidates-test problem); the largest and smallest values found across both searches are the absolute max and min.

Examples

On the closed disk $x^2+y^2\leq1$ , $f(x,y)=xy$ has interior critical value $f(0,0)=0$ , but the true absolute max $\tfrac12$ and min $-\tfrac12$ both occur on the boundary circle.