## Real Analysis Exchange

### Sets of Discontinuities for Functions Continuous on Flats

#### 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