On sets of discontinuities of functions continuous on all lines

Luděk Zajíček

Commentationes Mathematicae Universitatis Carolinae (2022)

  • Volume: 62 63, Issue: 4, page 487-505
  • ISSN: 0010-2628

Abstract

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Answering a question asked by K. C. Ciesielski and T. Glatzer in 2013, we construct a C 1 -smooth function f on [ 0 , 1 ] and a closed set M ⊂ graph f nowhere dense in graph f such that there does not exist any linearly continuous function on ℝ 2 (i.e., function continuous on all lines) which is discontinuous at each point of M . We substantially use a recent full characterization of sets of discontinuity points of linearly continuous functions on ℝ n proved by T. Banakh and O. Maslyuchenko in 2020. As an easy consequence of our result, we prove that the necessary condition for such sets of discontinuities proved by S. G. Slobodnik in 1976 is not sufficient. We also prove an analogue of this Slobodnik’s result in separable Banach spaces.

How to cite

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Zajíček, Luděk. "On sets of discontinuities of functions continuous on all lines." Commentationes Mathematicae Universitatis Carolinae 62 63.4 (2022): 487-505. <http://eudml.org/doc/299047>.

@article{Zajíček2022,
abstract = {Answering a question asked by K. C. Ciesielski and T. Glatzer in 2013, we construct a $C^1$-smooth function $f$ on $[0,1]$ and a closed set $M \subset \{\rm graph\} f$ nowhere dense in $\{\rm graph\} f$ such that there does not exist any linearly continuous function on $\{\mathbb \{R\}\}^2$ (i.e., function continuous on all lines) which is discontinuous at each point of $M$. We substantially use a recent full characterization of sets of discontinuity points of linearly continuous functions on $\{\mathbb \{R\}\}^n$ proved by T. Banakh and O. Maslyuchenko in 2020. As an easy consequence of our result, we prove that the necessary condition for such sets of discontinuities proved by S. G. Slobodnik in 1976 is not sufficient. We also prove an analogue of this Slobodnik’s result in separable Banach spaces.},
author = {Zajíček, Luděk},
journal = {Commentationes Mathematicae Universitatis Carolinae},
keywords = {linear continuity; discontinuity sets; Banach space},
language = {eng},
number = {4},
pages = {487-505},
publisher = {Charles University in Prague, Faculty of Mathematics and Physics},
title = {On sets of discontinuities of functions continuous on all lines},
url = {http://eudml.org/doc/299047},
volume = {62 63},
year = {2022},
}

TY - JOUR
AU - Zajíček, Luděk
TI - On sets of discontinuities of functions continuous on all lines
JO - Commentationes Mathematicae Universitatis Carolinae
PY - 2022
PB - Charles University in Prague, Faculty of Mathematics and Physics
VL - 62 63
IS - 4
SP - 487
EP - 505
AB - Answering a question asked by K. C. Ciesielski and T. Glatzer in 2013, we construct a $C^1$-smooth function $f$ on $[0,1]$ and a closed set $M \subset {\rm graph} f$ nowhere dense in ${\rm graph} f$ such that there does not exist any linearly continuous function on ${\mathbb {R}}^2$ (i.e., function continuous on all lines) which is discontinuous at each point of $M$. We substantially use a recent full characterization of sets of discontinuity points of linearly continuous functions on ${\mathbb {R}}^n$ proved by T. Banakh and O. Maslyuchenko in 2020. As an easy consequence of our result, we prove that the necessary condition for such sets of discontinuities proved by S. G. Slobodnik in 1976 is not sufficient. We also prove an analogue of this Slobodnik’s result in separable Banach spaces.
LA - eng
KW - linear continuity; discontinuity sets; Banach space
UR - http://eudml.org/doc/299047
ER -

References

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  8. Kuratowski K., Topology. Vol. I, Academic Press, New York, Państwowe Wydawnictwo Naukowe, Warszaw, 1966. 
  9. Lebesgue H., Sur les fonctions représentable analytiquement, J. Math. Pure Appl. (6) 1 (1905), 139–212 (French). 
  10. Shnol' É. É., Functions of two variables that are continuous along straight lines, Mat. Zametki 62 (1997), no. 2, 306–311 (Russian); translation in Math. Notes 62 (1997), no. 1–2, 255–259. MR1619865
  11. Slobodnik S. G., Expanding system of linearly closed sets, Mat. Zametki 19 (1976), no. 1, 67–84 (Russian); translation in Math. Notes 19 (1976), no. 1, 39–48. MR0409742
  12. Young W. H., Young G. C., Discontinuous functions continuous with respect to every straight line, Quart. J. Math. Oxford Series 41 (1910), 87–93. 
  13. Zajíček L., A remark on functions continuous on all lines, Comment. Math. Univ. Carolin. 60 (2019), no. 2, 211–218. MR3982469

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