Article
| Title: | On sets of discontinuities of functions continuous on all lines (English) |
| Author: | Zajíček, Luděk |
| Language: | English |
| Journal: | Commentationes Mathematicae Universitatis Carolinae |
| ISSN: | 0010-2628 (print) |
| ISSN: | 1213-7243 (online) |
| Volume: | 63 |
| Issue: | 4 |
| Year: | 2022 |
| Pages: | 487-505 |
| Summary lang: | English |
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| Category: | math |
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| Summary: | 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. (English) |
| Keyword: | linear continuity |
| Keyword: | discontinuity sets |
| Keyword: | Banach space |
| MSC: | 26B05 |
| MSC: | 46B99 |
| idZBL: | Zbl 07729555 |
| idMR: | MR4577043 |
| DOI: | 10.14712/1213-7243.2023.007 |
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| Date available: | 2023-04-20T13:55:16Z |
| Last updated: | 2023-10-27 |
| Stable URL: | http://hdl.handle.net/10338.dmlcz/151648 |
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| Reference: | [1] Banakh T., Maslyuchenko O.: Linearly continuous functions and $F_\sigma$-measurability.Eur. J. Math. 6 (2020), no. 1, 37–52. MR 4071455, 10.1007/s40879-019-00385-w |
| Reference: | [2] Bruckner A. M.: Differentiation of Real Functions.CRM Monograph Series, 5, American Mathematical Society, Providence, 1994. MR 1274044 |
| Reference: | [3] Ciesielski K. Ch., Glatzer T.: Sets of discontinuities of linearly continuous functions.Real Anal. Exchange 38 (2012/13), no. 2, 377–389. MR 3261883 |
| Reference: | [4] Ciesielski K. Ch., Glatzer T.: Sets of discontinuities for functions continuous on flats.Real Anal. Exchange 39 (2013/14), no. 1, 117–138. MR 3261903 |
| Reference: | [5] Ciesielski K. C., Miller D.: A continuous tale on continuous and separately continuous functions.Real Anal. Exchange 41 (2016), no. 1, 19–54. MR 3511935 |
| Reference: | [6] Dugundji J.: Topology.Allyn and Bacon, Boston, 1966. Zbl 0397.54003, MR 0193606 |
| Reference: | [7] Kershner R.: The continuity of functions of many variables.Trans. Amer. Math. Soc. 53 (1943), 83–100. MR 0007522, 10.1090/S0002-9947-1943-0007522-5 |
| Reference: | [8] Kuratowski K.: Topology. Vol. I.Academic Press, New York, Państwowe Wydawnictwo Naukowe, Warszaw, 1966. |
| Reference: | [9] Lebesgue H.: Sur les fonctions représentable analytiquement.J. Math. Pure Appl. (6) 1 (1905), 139–212 (French). |
| Reference: | [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. MR 1619865 |
| Reference: | [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. MR 0409742 |
| Reference: | [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. |
| Reference: | [13] Zajíček L.: A remark on functions continuous on all lines.Comment. Math. Univ. Carolin. 60 (2019), no. 2, 211–218. MR 3982469 |
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