Separately continuous functions with respect to a variable frame - Herasymchuk, V. H.; Maslyuchenko, V. K.; Maslyuchenko, O. V. (Ukrains’kyi Matematychnyi Zhurnal, 2004)
We show that the set D(f) of discontinuity points of a function f : R 2 → R continuous at every point p with respect to two variable linearly independent directions e 1(p) and e 2(p) is a set of the first category. Furthermore, if f is differentiable along one of directions, then D(f) is a nowhere dense set.