Open Access
2013/2014 Sets of Discontinuities for Functions Continuous on Flats
Krzysztof Chris Ciesielski, Timothy Glatzer
Real Anal. Exchange 39(1): 117-138 (2013/2014).


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.


Download Citation

Krzysztof Chris Ciesielski. Timothy Glatzer. "Sets of Discontinuities for Functions Continuous on Flats." Real Anal. Exchange 39 (1) 117 - 138, 2013/2014.


Published: 2013/2014
First available in Project Euclid: 1 July 2014

zbMATH: 1300.26008
MathSciNet: MR3261903

Primary: 26B05
Secondary: 58C05 , 58C07

Keywords: discontinuity sets , linear continuity , separate continuity

Rights: Copyright © 2013 Michigan State University Press

Vol.39 • No. 1 • 2013/2014
Back to Top