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

Abstract

The aim of this work is to extend the recent work of the author on the discrete frequency function to themore delicate continuous frequency function T , and further to investigate its relations to the Hardy–Littlewood maximalfunction M, and to the Lebesgue points. We surmount the intricate issue of measurability of T f by approaching itwith a sequence of carefully constructed auxiliary functions for which measurability is easier to prove. After this, wegive analogues of the recent results on the discrete frequency function. We then connect the points of discontinuity ofMf for f simple to the zeros of T f , and to the non-Lebesgue points of f .