Lebesgue Covering Theorem and level sets of continuous functions

We formulate and prove a dimension-theoretic generalization of a version of the Lebesgue Covering Theorem.

A generalized $n$-dimensional version of the Steinhaus Chessboard Theorem, recently proved algorithmically by Turzański and Ziajor, is a particular case of this result.

Moreover, we study two types of sets associated with a continuous function $g \colon [0,1]^n \to \mathbb{R}$.

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 $[0,1]^n$, 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 $[0, 1]^n$.