An optimal Brouwer's fixed point theorem for discontinuous functions

TL;DR

This paper improves upon Klee's 1961 results by establishing optimal bounds for fixed points of discontinuous functions in finite-dimensional Euclidean spaces, extending Brouwer's fixed point theorem.

Contribution

It provides the best possible bounds for fixed points of bounded-discontinuity functions, extending Brouwer's theorem to a broader class of functions.

Findings

Established optimal bounds for fixed points of discontinuous functions

Extended Brouwer's fixed point theorem to functions with bounded discontinuities

Proved the bounds are the best possible in finite-dimensional spaces

Abstract

Brouwer's fixed point theorem states that any continuous function from a closed $n$-dimensional ball to itself has a fixed point. In 1961, Klee showed that if such a function has discontinuities that are bounded, then it has a point that is close to being fixed. We improve upon Klee's results in any finite-dimensional Euclidean space, and prove that our bounds are the best possible.