Abstract
For a functionf(x, y), the setsJ a of all its discontinuity points with a jump ofa or more (that is, such that the oscillation of the function in the neighborhood of any point fromJ a is not smaller thana) are studied. Two cases are considered: (1)f is continuous along any straight line; (2)f is continuous along lines parallel to thex- andy-axes. In the first case, conditions that must be met by the setJ a are given. In the second case, it is shown that a (closed) setF can be the setJ a for a certain function if and only if the projections ofF on the coordinate axes nowhere dense.
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Mathematical Encyclopedia (I. M. Vinogradov, editor) [in Russian], Vol. 4, Soviet Encyclopedia, Moscow (1984).
Mathematical Encyclopedia (I. M. Vinogradov, editor) [in Russian], Vol. 5, Soviet Encyclopedia, Moscow (1985).
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Translated fromMatematicheskie Zametki, Vol. 62, No. 2, pp. 306–311, August, 1997.
Translated by V. N. Dubrovsky
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Shnol', E.E. Functions of two variables continuous along straigt lines. Math Notes 62, 255–259 (1997). https://doi.org/10.1007/BF02355912
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DOI: https://doi.org/10.1007/BF02355912