Chessboard and level sets of continuous functions

We show the following result: Let f Iⁿ → Rⁿ⁻¹ be a continuous function. Then, there exist p ∈ Rⁿ⁻¹ and compact subset S ⊂ f⁻¹[\] which connects some opposite faces of the n-dimensional unit cube Iⁿ. 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.