A Continuous Tale

Abstract

Assume that you have developed a good set of tools allowing you to decide which real functions of one real variable, $f\colon\mathbb{R}\to\mathbb{R}$, are continuous. (or, even more general, what are the continuity points of $f$).

(Q): How can such a tool-box be utilized to decide on the continuity of the functions $g\colon\mathbb{R}^n\to\mathbb{R}$ of $n$ real variables?

This is one of the questions which must be faced by any student taking multivariable calculus. Of course, such a student is following the footsteps of many generations of mathematicians, which were, and still are, struggling with the same general question. The aim of this article is to present the history and the current research related to this subject in a real analysis perspective, rather than in a more general, topological perspective.

In addition to surveying the results published so far, this exposition also includes several original results, as well as some new simplified versions of the proofs of older results. We also recall several intriguing open problems.