The Fundamental Theorem of Integral Calculus: a Volterra's generalization applied to flat functions

TL;DR

This paper extends the Fundamental Theorem of Calculus using Volterra's generalization, providing a weaker integral condition and offering new insights into flat functions and their properties.

Contribution

It replaces the classical theorem with Volterra's weaker form, broadening the theoretical framework and educational understanding of integral calculus for flat functions.

Findings

Replaces Riemann integral with Volterra's integral in fundamental theorem

Provides new proof techniques for flat functions with vanishing derivatives

Highlights the educational value of Volterra's integral in calculus

Abstract

In a recent paper [5] a smooth function f : [0; 1] --> R with all derivatives vanishing at 0 has been considered and a global condition, showing that f is indeed identically 0, has been presented. The purpose of this note is to replace the classical Fundamental Theorem of Calculus for the Riemann integral, as it has been used in [5], with a weaker form going back to Volterra [7], which is little known. Therefore the proof we propose in this paper turns to be important also from the teaching point of view, as long as in literature there are very few examples in which explicitly the lower integral and the upper integral of a function appear (usually the assumption that the function is Riemann-integrable is required).