Uniformity and self-neglecting functions

TL;DR

This paper extends Bloom's uniform convergence theorem for Beurling slowly varying functions by relaxing the continuity requirement, assuming only the Darboux property and measurability or Baire property.

Contribution

It introduces a broader class of functions for which Bloom's theorem holds, relaxing the continuity condition to the Darboux property and measurability or Baire property.

Findings

Results for  measurable functions with Darboux property

Extension to functions with Baire property

Relaxation of continuity assumption in Bloom's theorem

Abstract

We relax the continuity assumption in Bloom's uniform convergence theorem for Beurling slowly varying functions \phi. We assume that \phi has the Darboux property, and obtain results for \phi measurable or having the Baire property.