A Remark on Lebesgue Criterion

TL;DR

This paper presents a new proof linking Lebesgue differentiation and Riemann integrability, showing that functions with limits everywhere are almost everywhere continuous, thus deepening the understanding of the relationship between Lebesgue and Riemann integrals.

Contribution

It introduces a novel proof connecting Lebesgue differentiation theorem with the Lebesgue criterion for Riemann integrability, offering new insights into function continuity.

Findings

Every Riemann-integrable function with limit everywhere on the interior is almost everywhere continuous.

The proof establishes a new connection between Lebesgue differentiation theorem and Lebesgue criterion.

Provides a different perspective on the relationship between Riemann and Lebesgue integrability.

Abstract

We remark a variant of the existence part of the fundamental theorem of calculus, which, together with the Lebesgue differentiation theorem, constitute a new proof that every Riemann-integrable function on a compact interval having limit everywhere on the interior is almost everywhere continuous with respect to Lebesgue measure. The proof is intended as a new connection between Lebesgue differentiation theorem and Lebesgue criterion of Riemann integrability.