Continuity and discreteness are often seen as two opposing concepts, but we can use discrete combinatorics to prove many properties of continuous functions. The main benefit is that we can easily (at least when the dimension is not too large) imagine and also draw some pictures of proof. This works particularly well when we try to prove a theorem on the existence of special points of a function, for example, a fixed point or a zero of a function. One example of this sort of theorem is Poincaré-Miranda’s theorem which is proved in this work by discrete combinatorics on vertices of a triangulation of a cube and Sperner’s lemma. We can show that Poincaré-Miranda’s theorem implies the domain invariance theorem. In the end, we derive a simple but important corollary, the dimension invariance theorem.
|
