A fixed point theorem for discontinuous functions

We establish the following fixed point theorem. Let P be a non-empty polytope in the n-dimensional Euclidean space and let f be a function from P to P satisfying that for every x in P for which f(x) 6= x there exists a neighborhood of x in P such that for any y and z in this neighborhood the inner product of f(y)− y and f(z)− z is non-negative. Then there exists x∗ ∈ P satisfying f(x∗) = x∗, i.e., x∗ is a fixed point of the function f . This condition allows for various kinds of discontinuities and irregularities of the underlying function and contains Brouwer’s fixed point theorem for continuous functions as a special case. Moreover, a function satisfying the condition may be neither lower nor upper semicontinuous. We also discuss the application of the result in the area of noncooperative game theory.