Topics in uniform continuity

TL;DR

This paper reviews various generalizations of uniform continuity, discusses open problems, and explores metric spaces where all continuous functions exhibit near-uniform continuity, broadening understanding of function behavior in different spaces.

Contribution

It compiles existing results and open problems on generalized uniform continuity and introduces classes of metric spaces with near-uniform continuity properties.

Findings

Identification of classes of functions generalizing uniform continuity

Discussion of metric spaces where all continuous functions are close to uniformly continuous

Presentation of open problems in the area

Abstract

This paper collects results and open problems concerning several classes of functions that generalize uniform continuity in various ways, including those metric spaces (generalizing Atsuji spaces) where all continuous functions have the property of being close to uniformly continuous.