Sets of Discontinuities for Functions Continuous on Flats
Krzysztof Chris Ciesielski, Timothy Glatzer · Real Analysis Exchange · 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.