Real Analysis Exchange

A Continuous Tale

Krzysztof Chris Ciesielski and David Miller

Full-text: Access denied (no subscription detected)

We're sorry, but we are unable to provide you with the full text of this article because we are not able to identify you as a subscriber. If you have a personal subscription to this journal, then please login. If you are already logged in, then you may need to update your profile to register your subscription. Read more about accessing full-text

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.

Article information

Source
Real Anal. Exchange, Volume 41, Number 1 (2016), 19-54.

Dates
First available in Project Euclid: 29 March 2017

Permanent link to this document
https://projecteuclid.org/euclid.rae/1490752816

Mathematical Reviews number (MathSciNet)
MR3511935

Zentralblatt MATH identifier
06848915

Subjects
Primary: 26B05: Continuity and differentiation questions
Secondary: 54C05: Continuous maps 54C08: Weak and generalized continuity

Keywords
separate continuity linear continuity

Citation

Ciesielski, Krzysztof Chris; Miller, David. A Continuous Tale. Real Anal. Exchange 41 (2016), no. 1, 19--54. https://projecteuclid.org/euclid.rae/1490752816


Export citation