Sets of Discontinuities of Linearly Continuous Functions

Authors

Krzysztof Ciesielski, West Virginia UniversityFollow

Document Type

Article

Publication Date

2013

College/Unit

Eberly College of Arts and Sciences

Department/Program/Center

Mathematics

Abstract

The class of linearly continuous functions f:Rⁿ-->R, that is, having continuous restrictions f|L to every straight line L, have been studied since the dawn of the twentieth century. In this paper we refine a description of the form that the sets D(f) of points of discontinuities of such functions can have. It has been proved by Slobodnik that D(f) must be a countable union of isometric copies of the graphs of Lipschitz functions h:K-->R, where K is a compact nowhere dense subset of R^n-1. Since the class Dⁿ of all sets D(f), with f:Rⁿ-->R being linearly continuous, is evidently closed under countable unions as well as under isometric images, the structure of Dⁿ will be fully discerned upon deciding precisely which graphs of the Lipschitz functions h:K-->R, K being compact nowhere dense subset of R^n-1, belong to Dⁿ. Towards this goal, we prove that Dⁿ contains the graph of any such h:K-->R whenever h is a restriction of convex function from R^n-1 into R. Moreover, for n=2, D² contains the graph of any such h, if h can be extended to a C² function H:R-->R. At the same time, we provide an example, showing that this last result need not hold when H is just differentiable with bounded derivative (so Lipschitz).

Digital Commons Citation

Ciesielski, Krzysztof, "Sets of Discontinuities of Linearly Continuous Functions" (2013). Faculty & Staff Scholarship. 846.
https://researchrepository.wvu.edu/faculty_publications/846