Some classical analysis results for continuous definable mappings

TL;DR

This paper extends fundamental theorems from smooth mappings to continuous definable mappings within o-minimal structures, highlighting their natural properties and broader applicability.

Contribution

It provides the first systematic extension of key smooth mapping results to continuous definable mappings in o-minimal structures.

Findings

Key theorems like Brouwer degree, inverse function, and Sard's theorem extend to definable mappings.

Definable mappings exhibit properties enabling classical analysis results to hold.

The approach relies on the geometric and topological properties of definable sets.

Abstract

In this paper, we show that some fundamental results for smooth mappings (e.g., the Brouwer degree formula, the implicit function and inverse function theorems, the mean value theorem, Sard's theorem, Hadamard's global invertibility criteria, Pourciau's surjectivity and openness results) have natural extensions for continuous mappings that are definable in o-minimal structures. The arguments rely on nice properties of definable mappings and sets.