multivariate function domain

For a (real) multivariate function (let’s say $z=f(x,y)$), its domain is a region $R\in {\mathbb{R}}^n$ where $n$ is $f$‘s arity.

The boundary of $R$ is a curve that separates points internal to the domain from the points outside the domain. Formally, a region in a plane is bounded if it lies inside a disk of finite radius. Note that boundary points can be either in or out of the domain, and two boundary points don’t have to all be in or out of the domain.

• The domain is closed if $R$ contains all boundary points, if any.
• The domain is open if $R$ contains no boundary points.
• In cases where there is no boundary (e.g. when the domain is the entire plane), then the domain is both open and closed.
• The domain can be neither open nor closed if $R$ contains some but not all boundary points.