Functions of two variables continuous along straight lines

For a function $f(x,y)$, the sets $J_a$ of all its discontinuity points with a jump of $a$ or more (that is, such that the oscillation of the function in the neighborhood of any point from $J_a$ is not smaller than $a$) are studied. Two cases are considered: (1) $f$ is continuous along any straight line; (2) $f$ is continuous along lines parallel to the $x$- and $y$-axes. In the first case, conditions that must be met by the set $J_a$ are given. In the second case, it is shown that a (closed) set $F$ can be the set $J_a$ for a certain function if and only if the projections of $F$ on the coordinate axes nowhere dense.