An optimal Brouwer's fixed point theorem for discontinuous functions

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.