Fibers of continuous functions that connect or separate some opposite faces of the unit cube, and a generalization of the Lebesgue Covering Theorem

We formulate and prove a dimension-theoretic generalization of the Lebesgue Covering Theorem. A generalized $n$-dimensional version of the Steinhaus Chessboard Theorem, recently proved by Turzański and Ziajor, is a simple consequence of this result. Moreover, we study two types of sets associated with a continuous function $g\text{\hspace{0.17em}⁣}:I^n\to \mathbb{R}$. Namely, the set of all points $p\in \mathbb{R}$ such that the fiber $g^{-1}[\left\{p\right\}]$ connects $i$th opposite faces of $I^n$, and the set of all points $p\in \mathbb{R}$ such that the fiber $g^{-1}[\left\{p\right\}]$ separates $i$th opposite faces of $I^n$. We provide necessary and sufficient conditions for the existence of a continuous function $g\text{\hspace{0.17em}⁣}:I^n\to \mathbb{R}$ in terms of these sets.