A new proof of Poincar\'e-Miranda theorem based on the classification of one-dimensional manifolds

TL;DR

This paper presents an elementary proof of the Poincaré-Miranda theorem using Sard's theorem and the classification of one-dimensional manifolds, offering a new perspective on this fundamental result.

Contribution

It introduces a novel proof approach for the Poincaré-Miranda theorem leveraging classical tools from differential topology.

Findings

Provides a simpler, more elementary proof of the theorem.

Connects the theorem to the classification of 1D manifolds.

Highlights the role of Sard's theorem in topological proofs.

Abstract

This note gives a new elementary proof of Poincar\'e-Miranda theorem based on Sard's theorem and the simple classification of one-dimensional manifolds.