Chessboard and level sets of continuous functions

Abstract

We provide the following result and its discrete equivalent: Let $f:I^n\to {\mathbb{R}}^{n-1}$ be a continuous function. Then, there exist a point $p\in {\mathbb{R}}^{n-1}$ and a compact subset $S\subset f^{-1}\left[\left\{p\right\}\right]$ which connects some opposite faces of the $n$-dimensional unit cube $I^n$. We give an example that shows it cannot be generalized to path-connected sets. Additionally, we show that a version of the Steinhaus Chessboard Theorem and the Brouwer Fixed Point Theorem are simple consequences of this result.