we define a function in to have compact support if there is some such that . Three questions:
1. Show that if all the are continuously differentiable functions, with at least one of them having compact support.
2. prove that if and are continuously differentiable functions in and at least one of them has compact support, then
3. Show that the coordinate functions provide an example of functions on (none with compact support) such that
where is the ball of radius about the origin in . The coordinate functions are defined as .