Functions of two variables continuous along straight lines (Q1277522)

Bounded functions on the square \(\{0\leq x\leq 1, 0\leq y\leq 1\}\) are considered. Denote: by \(C_k\) the class of functions \(f\) such that the restriction of \(f(x,y)\) to any straight line is a continuous function on the corresponding segment; by \(J_a(f)\) or simply \(J_a\), the (closed) set of discontinuity points of \(f\) with a jump of \(a\) or more (the oscillation at \(a\) is not less than \(a\)); by \(S_{xy}\) the class of functions continuous in each of the variables \(x\) and \(y\) separately. Theorem 1. A set \(F\) in the plane is the set \(J_1\) for a certain function of class \(C_k\) if and only if the following conditions hold: (1) the projection of \(F\) onto any line is a nowhere dense set; (2) the directions in which points of \(F\) are seen from any point constitute a nowhere dense set. Corollary 1: The set \(J(f)\) of \(f\) in \(C_k\) does not contain any connected subset. Corollary 2: The planar Lebesgue measure of \(J(f)\) is equal to zero. Theorem 2: A closed set \(F\) is the set \(J_a\) for a certain function \(f\) in \(C_{xy}\) if and only if the projections of \(F\) onto the coordinate axes are nowhere dense (by \(J(f)\) is denoted the set of discontinuity points of \(f\), i.e., the union of \(J_{1/n}(f)\) for \(n= 1,2,\dots\)). Many interesting examples are provided.