Sets of Discontinuities for Functions Continuous on Flats

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Eberly College of Arts and Sciences




For families F of flats (i.e., affine subspaces) of R n , we investigate the classes of F-continuous functions f : R n → R, whose restrictions f F are continuous for every F ∈ F. If Fk is the class of all k-dimensional flats, then F1-continuity is known as linear continuity; if F + k stands for all F ∈ Fk parallel to vector subspaces spanned by coordinate vectors, then F + 1 -continuous maps are the separately continuous functions, that is, those which are continuous in each variable separately. For the classes F = F + k , we give a full characterization of the collections D(F) of the sets of points of discontinuity of F-continuous functions. We provide the structural results on the families D(Fk) and give a full characterization of the collections D(Fk) in the case when k ≥ n/2. In particular, our characterization of the class D(F1) for R 2 solves a 60 year old problem of Kronrod.

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