The Implicit Function Theorem for continuous functions

TL;DR

This paper introduces a homological approach to the implicit function theorem and Darboux theorem for continuous maps on topological manifolds, extending classic results to less smooth functions.

Contribution

It develops a new homological framework for the implicit function theorem applicable to continuous functions, broadening the scope of classical differential topology.

Findings

Established a homological version of the implicit function theorem for continuous maps.

Extended versions of Darboux's theorem to less smooth differentiable maps.

Proved classical theorems under weaker smoothness assumptions.

Abstract

In the present paper we obtain a new homological version of the implicit function theorem and some versions of the Darboux theorem. Such results are proved for continuous maps on topological manifolds. As a consequence, some versions of these classic theorems are proved when we consider differenciable (not necessarily C^1) maps.