Mean value integral inequalities

TL;DR

This paper investigates conditions under which a function with a dense set of points where its derivative is zero must be constant, introducing new mean value theorems for integrals proven via Lebesgue and Riemann integration.

Contribution

It introduces new mean value theorems for integrals and applies them to characterize when functions with zero derivatives on dense sets are constant.

Findings

New mean value theorems for Lebesgue and Riemann integrals

Conditions ensuring functions with dense zero-derivative sets are constant

Elementary and Lebesgue-based proofs of the theorems

Abstract

Let $F:[a,b]\longrightarrow \R$ have zero derivative in a dense subset of $[a,b]$. What else we need to conclude that $F$ is constant in $[a,b]$? We prove a result in this direction using some new Mean Value Theorems for integrals which are the real core of this paper. These Mean Value Theorems are proven easily and concisely using Lebesgue integration, but we also provide alternative and elementary proofs to some of them which keep inside the scope of the Riemann integral.