To show that each of the hypotheses of Brouwer's theorem is necessary, find examples of functions f : S → S with S ⊆ R that do not have fixed points, where
1. f is continuous and S is convex but not compact
2. f is continuous and S is compact but not convex
3. S is compact and convex but f is not continuous
The following proposition, which is equivalent to Brouwer's theorem, asserts that it is impossible to map the unit ball continuously on to its boundary.