Sets of Discontinuities of Linearly Continuous Functions

The class of linearly continuous functions f:Rn--\u3eR, 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--\u3eR, where K is a compact nowhere dense subset of Rn-1. Since the class Dn of all sets D(f), with f:Rn--\u3eR 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--\u3eR, K being compact nowhere dense subset of Rn-1, belong to Dn. Towards this goal, we prove that Dn contains the graph of any such h:K--\u3eR whenever h is a restriction of convex function from Rn-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--\u3eR. 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)