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 Rn-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 & Unknown;Rn-1 being compact nowhere dense, belong to D-n. Towards this goal, we prove that D-2 contains the graph of any such h: K -> R whenever h can be extended to a C-2 function h : R -> R. Moreover, for every n > 1, D-n contains the graph of any h: K -> R, where K is closed nowhere dense in Rn-1 and h is a restriction of a convex function h: Rn-1 -> R. In addition, we provide an example, showing that the above mentioned result on C-2 functions need not hold when h is just differentiate with bounded derivative (so Lipschitz).
