# The frequency function and its connections to the Lebesgue points and the Hardy-Littlewood maximal function

## TL;DR

This paper extends the concept of the frequency function to the continuous case, explores its relation to the Hardy-Littlewood maximal function and Lebesgue points, and establishes connections between discontinuities, zeros, and non-Lebesgue points.

## Contribution

It introduces a continuous frequency function $ ext{T}$, proves its measurability, and links its properties to the Hardy-Littlewood maximal function and Lebesgue points.

## Key findings

- Connected discontinuities of $ ext{M}f$ to zeros of $ ext{T}f$
- Linked non-Lebesgue points of $f$ to zeros of $ ext{T}f$
- Extended results from discrete to continuous frequency functions

## Abstract

The aim of this work is to extend the recent work of the author on the discrete frequency function to the more delicate continuous frequency function $\mathcal{T}$, and further to investigate its relations to the Hardy-Littlewood maximal function $\mathcal{M}$, and to the Lebesgue points. We surmount the intricate issue of measurability of $\mathcal{T}f$ by approaching it with a sequence of carefully constructed auxiliary functions for which measurability is easier to prove. After this we give analogues of the recent results on the discrete frequency function. We then connect the points of discontinuity of $\mathcal{M}f$ for $f$ simple to the zeros of $\mathcal{T}f$, and to the non-Lebesgue points of $f$.