[Paper Review] On the Almost Everywhere Continuity

Joël Blot|arXiv (Cornell University)|Nov 13, 2014

Advanced Banach Space Theory3 citations

TL;DR

This paper establishes characterizations of Lebesgue-almost everywhere continuity for real-valued functions on a compact interval, showing that if a function has finite one-sided limits almost everywhere, then it is continuous almost everywhere. The key contribution is a sufficient condition for Riemann integrability based on the existence of finite one-sided limits almost everywhere.

ABSTRACT

The aim of this paper is to provide characterizations of the Lebesgue-almost everywhere continuity of a function f : [a, b] $ ightarrow$ R. These characterizations permit to obtain necessary and sufficient conditions for the Riemann integrability of f .

Motivation & Objective

• To characterize functions that are continuous almost everywhere with respect to Lebesgue measure.
• To establish necessary and sufficient conditions for Riemann integrability based on the existence of finite one-sided limits.
• To generalize results from Rémondière and Saada on discontinuity sets of regulated functions.
• To clarify the relationship between one-sided limits, oscillation, and continuity in the context of Lebesgue measure.
• To provide a framework for analyzing discontinuity sets using oscillation functions and measure-theoretic tools.

Proposed method

• Introduces left-hand and right-hand oscillation functions $\omega_L(x)$ and $\omega_R(x)$ to quantify discontinuity severity at a point.
• Uses the equivalence: $\omega_L(x) = 0$ iff $f$ is left-continuous at $x$, and similarly for $\omega_R(x)$.
• Applies measure-theoretic arguments: if a set has positive measure, it cannot be uncountably disjoint union of positive-measure intervals.
• Employs a key lemma showing that if $f$ has finite right-hand limit at all but a negligible set, then $\omega_L(x) = 0$ a.e.
• Proves that the set of discontinuity points is Lebesgue-negligible (or at most countable) if one-sided limits exist finitely a.e.
• Combines results on left and right continuity via union of negligible sets to conclude a.e. continuity.

Experimental results

Research questions

• RQ1 Under what conditions is a bounded function on $[a,b]$ Riemann integrable based on one-sided limit behavior?
• RQ2 When does the existence of finite one-sided limits almost everywhere imply almost everywhere continuity?
• RQ3 How do oscillation functions $\omega_L$ and $\omega_R$ relate to the structure of discontinuity sets?
• RQ4 Can the discontinuity set of a function be characterized purely by the existence of one-sided limits?
• RQ5 What is the precise relationship between the size of the discontinuity set and the existence of finite one-sided limits?

Key findings

• If $f$ has finite right-hand limits at all points of $[a,b)$ except on a Lebesgue-negligible set, then $f$ is continuous almost everywhere on $[a,b]$.
• The set of points where $f$ is discontinuous is Lebesgue-negligible if and only if the set of points where $f$ fails to have a finite left-hand or right-hand limit is Lebesgue-negligible.
• For bounded functions, Riemann integrability is equivalent to the set of discontinuity points being Lebesgue-negligible, which is equivalent to the existence of finite one-sided limits a.e.
• If $f$ is right-hand continuous or left-hand continuous on $[a,b]$, then its discontinuity set is at most countable, and hence $f$ is Riemann integrable if bounded.
• The oscillation functions $\omega_L$ and $\omega_R$ provide a precise measure of discontinuity, with $\omega_L(x) = 0$ iff $f$ is left-continuous at $x$.
• The union of two Lebesgue-negligible sets (e.g., where $\omega_L > 0$ and $\omega_R > 0$) remains Lebesgue-negligible, ensuring a.e. continuity under the stated conditions.