Past exam of the mathematics course of the University of Cambridge / 2015 / ii / Paper 1 / 2I / i / Solution

The closure is closed and bounded in ${\mathbb{R}}^2$, hence compact by the Heine-Borel theorem. It is nonempty. The extreme value theorem therefore gives a point of the closure at which the continuous function attains its maximum. The closure, rather than the open domain, is essential.