An optimal Brouwer's fixed point theorem for discontinuous functions
Metric Geometry
2025-12-18 v1
Abstract
Brouwer's fixed point theorem states that any continuous function from a closed -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.
Cite
@article{arxiv.2512.14934,
title = {An optimal Brouwer's fixed point theorem for discontinuous functions},
author = {Henry Adams and Florian Frick},
journal= {arXiv preprint arXiv:2512.14934},
year = {2025}
}