A Fixed Point Theorem for Discontinuous Functions

In this paper we prove the following fixed point theorem. Consider a non-empty bounded polyhedron P and a function f from P to P such that for every x in P with f(x) not equal to x there is a ball in P around x such that the inner product of f(y)-y and f(z)-z is nonnegative for any two elements y and z in the ball. Then f has a fixed point. The condition allows for various discontinuities and irregularities of the function. In case f is a continuous function, the condition is automatically satisfied and thus the Brouwer fixed point theorem is implied by the result. We illustrate that a function that satisfies the condition is not necessarily upper or lower semi-continuous. A game-theoretic application is also discussed