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Generalized Lagrange Theorem

Karolina Zając

TL;DR

The paper generalizes the Mean Value Theorem to continuous, potentially non-differentiable functions in several variables by introducing the bisequential tangent cone (BTC) as a replacement for the derivative. It develops Rolle- and Lagrange-type results via BTC, analyzes normal cones to obtain a geometric mean-value condition, and connects BTC with the Clarke subdifferential in the Lipschitz setting, showing their equivalence and providing a geometric interpretation of Lebourg's theorem. Together, these results extend nonsmooth analysis tools beyond classical derivatives and offer a cohesive geometric framework for approximating tangential behavior of continuous maps.

Abstract

The present paper is devoted to possible generalizations of the classic Lagrange Mean Value Theorem. We consider a real-valued function of several variables that is only assumed to be continuous. The main concept is to replace the notion of the derivative by the so called bisequential tangent cone. We first prove Rolle and Lagrange type results and then we turn to comparing this cone with the Clarke subdifferential in the case of a Lipschitz function. We also investigate an approach using normal cones.

Generalized Lagrange Theorem

TL;DR

The paper generalizes the Mean Value Theorem to continuous, potentially non-differentiable functions in several variables by introducing the bisequential tangent cone (BTC) as a replacement for the derivative. It develops Rolle- and Lagrange-type results via BTC, analyzes normal cones to obtain a geometric mean-value condition, and connects BTC with the Clarke subdifferential in the Lipschitz setting, showing their equivalence and providing a geometric interpretation of Lebourg's theorem. Together, these results extend nonsmooth analysis tools beyond classical derivatives and offer a cohesive geometric framework for approximating tangential behavior of continuous maps.

Abstract

The present paper is devoted to possible generalizations of the classic Lagrange Mean Value Theorem. We consider a real-valued function of several variables that is only assumed to be continuous. The main concept is to replace the notion of the derivative by the so called bisequential tangent cone. We first prove Rolle and Lagrange type results and then we turn to comparing this cone with the Clarke subdifferential in the case of a Lipschitz function. We also investigate an approach using normal cones.
Paper Structure (3 sections, 14 theorems, 66 equations)

This paper contains 3 sections, 14 theorems, 66 equations.

Table of Contents

  1. Bisequential tangent cones
  2. Normal cones
  3. Clarke subdifferential

Key Result

Theorem 1.1

Let $a,b \in \mathbb{R}, a<b$ and let $f:[a,b] \rightarrow \mathbb{R}$ be a continuous function, differentiable on an open interval $(a,b)$. Then

Theorems & Definitions (40)

  • Theorem 1.1: Lagrange
  • Definition 1.2
  • Definition 1.3
  • Remark 1.4
  • Definition 1.5
  • Remark 1.6
  • Remark 1.7
  • Remark 1.8
  • Proposition 1.9
  • proof
  • ...and 30 more