Real Analysis Exchange

Sets of Discontinuities for Functions Continuous on Flats

Krzysztof Chris Ciesielski and Timothy Glatzer

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Abstract

For families \(\mathcal{F}\) of flats (i.e., affine subspaces) of \(\mathbb{R}^n\), we investigate the classes of \(\mathcal{F}\)-continuous functions \(f\colon\mathbb{R}^n\to\mathbb{R}\), whose restrictions \(f\restriction F\) are continuous for every \(F\in\F\). If \(\mathcal{F}_k\) is the class of all \(k\)-dimensional flats, then \(\mathcal{F}_1\)-continuity is known as linear continuity; if \(\mathcal{F}_k^+\) stands for all \(F\in\mathcal{F}_k\) parallel to vector subspaces spanned by coordinate vectors, then \(\mathcal{F}_1^+\)-continuous maps are the separately continuous functions, that is, those which are continuous in each variable separately. For the classes \(\mathcal{F}=\mathcal{F}_k^+\), we give a full characterization of the collections \(\mathcal{D}\mathcal{F}(\mathcal{F})\) of the sets of points of discontinuity of \(\F\)-continuous functions. We provide the structural results on the families \(\mathcal{D}(\mathcal{F}_k)\) and give a full characterization of the collections \(\mathcal{D}(\mathcal{F}_k)\) in the case when \(k\geq n/2\). In particular, our characterization of the class \(\mathcal{D}(\mathcal{F}_1)\) for \(\mathbb{R}^2\) solves a 60 year old problem of Kronrod.

Article information

Source
Real Anal. Exchange, Volume 39, Number 1 (2013), 117-138.

Dates
First available in Project Euclid: 1 July 2014

Permanent link to this document
https://projecteuclid.org/euclid.rae/1404230144

Mathematical Reviews number (MathSciNet)
MR3261903

Zentralblatt MATH identifier
1300.26008

Subjects
Primary: 26B05: Continuity and differentiation questions
Secondary: 58C07: Continuity properties of mappings 58C05: Real-valued functions

Keywords
separate continuity linear continuity discontinuity sets

Citation

Ciesielski, Krzysztof Chris; Glatzer, Timothy. Sets of Discontinuities for Functions Continuous on Flats. Real Anal. Exchange 39 (2013), no. 1, 117--138. https://projecteuclid.org/euclid.rae/1404230144


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