Chessboard and level sets of continuous functions

Abstract

We show the following result: Let $f \colon I^n \to \mathbb{R}^{n-1}$ be a continuous function. Then, there exist $p \in \mathbb{R}^{n-1}$ and 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. We also provide a discrete version of this result which is inspired by the $n$-dimensional Steinhaus Chessboard Theorem. Additionally, we show that the latter one and the Brouwer Fixed Point Theorem are simple consequences of the main result.