#### 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:**R**^{n}-->**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 **D**^{n} of all sets D(f), with f:**R**^{n}-->**R** being linearly continuous, is evidently closed under countable unions as well as under isometric images, the structure of **D**^{n} 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 **D**^{n}. Towards this goal, we prove that **D**^{n} 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, **D**^{2} contains the graph of any such h, if h can be extended to a **C**^{2} 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