The following proof of the Fixpoint Theorem of Brouwer was outlined as classroom exercise at the University of Montpellier in the early sixties.
a. Reduction to the -differentiable case. Let be the function . Let given by: on and equal to in the complement of , where the constant is chosen such that the integral of over equals . Let be continuous. Let be the extension of to , for which we have on . For put . Show that the restriction of to maps into . Show that the mappings are continuously differentiable and approximate in the topology of uniform convergence the mapping . Show that if there exists a continuous mapping without fixpoints, then there will also exist a continuously differential map without fixpoints. It follows, that it suffices to proof the Brouwer Fixpoint Theorem only for continuously differentiable mappings.
b. Proof for -differentiable mappings. The proof is by contradiction.
Assume, that the continuously differential mapping has no
the mapping, such that for every point
the points are in that order on a line of .
The mapping is also continuously differentiable and satisfies for
. We write
The following is a contradiction: