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Mat. Zametki, 1997, Volume 62, Issue 2, Pages 306–311 (Mi mz1612)  

This article is cited in 1 scientific paper (total in 3 paper)

Functions of two variables continuous along straight lines

È. È. Shnol'

Institute of Mathematical Problems of Biology, Russian Academy of Sciences

Abstract: 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.

DOI: https://doi.org/10.4213/mzm1612

Full text: PDF file (145 kB)
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English version:
Mathematical Notes, 1997, 62:2, 255–259

Bibliographic databases:

UDC: 517.51
Received: 14.04.1995
Revised: 10.10.1996

Citation: È. È. Shnol', “Functions of two variables continuous along straight lines”, Mat. Zametki, 62:2 (1997), 306–311; Math. Notes, 62:2 (1997), 255–259

Citation in format AMSBIB
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\by \`E.~\`E.~Shnol'
\paper Functions of two variables continuous along straight lines
\jour Mat. Zametki
\yr 1997
\vol 62
\issue 2
\pages 306--311
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\crossref{https://doi.org/10.4213/mzm1612}
\mathscinet{http://www.ams.org/mathscinet-getitem?mr=1619865}
\zmath{https://zbmath.org/?q=an:0915.26004}
\transl
\jour Math. Notes
\yr 1997
\vol 62
\issue 2
\pages 255--259
\crossref{https://doi.org/10.1007/BF02355912}
\isi{http://gateway.isiknowledge.com/gateway/Gateway.cgi?GWVersion=2&SrcApp=PARTNER_APP&SrcAuth=LinksAMR&DestLinkType=FullRecord&DestApp=ALL_WOS&KeyUT=000071268600031}


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    Citing articles on Google Scholar: Russian citations, English citations
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    This publication is cited in the following articles:
    1. V. I. Arnol'd, R. M. Borisyuk, I. M. Gel'fand, Yu. S. Ilyashenko, V. Yu. Lunin, E. V. Nikolaev, Yu. B. Radvogin, M. A. Roitberg, Ya. G. Sinai, A. I. Khibnik, “Emmanuil El'evich Shnol' (on his 70th birthday)”, Russian Math. Surveys, 54:3 (1999), 677–683  mathnet  crossref  crossref  mathscinet  zmath  adsnasa  isi
    2. Ciesielski K.Ch., Miller D., “A Continuous Tale on Continuous and Separately Continuous Functions”, Real Anal. Exch., 41:1 (2016), 19–54  crossref  mathscinet  zmath  isi
    3. A. I. Aptekarev, A. L. Afendikov, F. I. Ataullakhanov, N. K. Balabaev, V. N. Biktashev, I. V. Biktasheva, R. M. Borisyuk, N. D. Vvedenskaya, R. D. Dagkesamanskii, Yu. G. Zarkhin, Yu. S. Ilyashenko, V. D. Lakhno, V. Yu. Lunin, N. L. Lunina, E. V. Nikolaev, V. S. Posvyanskii, M. A. Roitberg, V. S. Ryaben'kii, L. B. Ryashko, Ya. G. Sinai, V. M. Tikhomirov, A. A. Tokarev, A. G. Urzhumtsev, A. I. Khibnik, “To the memory of Èmmanuil Èl'evich Shnol'”, Russian Math. Surveys, 72:1 (2017), 185–198  mathnet  crossref  crossref  mathscinet  adsnasa  isi  elib
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