Geodesic
Functions of two variables continuous along straight lines
Matematičeskie zametki, Tome 62 (1997) no. 2, pp. 306-311.

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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. 
@article{MZM_1997_62_2_a11,
     author = {\`E. \`E. Shnol'},
     title = {Functions of two variables continuous along straight lines},
     journal = {Matemati\v{c}eskie zametki},
     pages = {306--311},
     publisher = {mathdoc},
     volume = {62},
     number = {2},
     year = {1997},
     language = {ru},
     url = {http://geodesic.mathdoc.fr/item/MZM_1997_62_2_a11/}
}
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È. È. Shnol'. Functions of two variables continuous along straight lines. Matematičeskie zametki, Tome 62 (1997) no. 2, pp. 306-311. http://geodesic.mathdoc.fr/item/MZM_1997_62_2_a11/

[1] Matematicheskaya entsiklopediya, T. 4, ed. Vinogradov I. M., Sovetskaya entsiklopediya, M., 1984

[2] Matematicheskaya entsiklopediya, T. 5, ed. Vinogradov I. M., Sovetskaya entsiklopediya, M., 1985