We provide the following result and its discrete equivalent: Let <InlineEquation ID="IEq1"> <EquationSource Format="TEX">$f :I^n \rightarrow \mathbb {R}^{n-1}$</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <mi>f</mi> <mo>:</mo> <msup> <mi>I</mi> <mi>n</mi> </msup> <mo stretchy="false">→</mo> <msup> <mrow> <mi mathvariant="double-struck">R</mi> </mrow> <mrow> <mi>n</mi> <mo>-</mo> <mn>1</mn> </mrow> </msup> </mrow> </math></EquationSource> </InlineEquation> be a continuous function. Then, there exist a point <InlineEquation ID="IEq2"> <EquationSource Format="TEX">$p \in \mathbb {R}^{n-1}$</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <mi>p</mi> <mo>∈</mo> <msup> <mrow> <mi mathvariant="double-struck">R</mi> </mrow> <mrow> <mi>n</mi> <mo>-</mo> <mn>1</mn> </mrow> </msup> </mrow> </math></EquationSource> </InlineEquation> and a compact subset <InlineEquation ID="IEq3"> <EquationSource Format="TEX">$S \subset f^{-1}\left[ \left\{ p\right\} \right] $</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <mi>S</mi> <mo>⊂</mo> <msup> <mi>f</mi> <mrow> <mo>-</mo> <mn>1</mn> </mrow> </msup> <mfenced close="]" open="["> <mfenced close="}" open="{"> <mi>p</mi> </mfenced> </mfenced> </mrow> </math></EquationSource> </InlineEquation> which connects some opposite faces of the n-dimensional unit cube <InlineEquation ID="IEq4"> <EquationSource Format="TEX">$I^n$</EquationSource> <EquationSource Format="MATHML"><math> <msup> <mi>I</mi> <mi>n</mi> </msup> </math></EquationSource> </InlineEquation>. 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.

Chessboard and Level Sets of Continuous Functions

期刊论文
发表日期：

2026年6月3日

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Michał Dybowski,
Przemysław Górka

摘要

We provide the following result and its discrete equivalent: Let $f :I^n \rightarrow \mathbb {R}^{n-1}$ $f : I^{n} \to R^{n - 1}$ be a continuous function. Then, there exist a point $p \in \mathbb {R}^{n-1}$ $p \in R^{n - 1}$ and a compact subset $S \subset f^{-1}\left[ \left\{ p\right\} \right] $ $S \subset f^{- 1} [\{p\}]$ which connects some opposite faces of the n-dimensional unit cube $I^n$ $I^{n}$ . 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.