| Title:
|
A remark on 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:
|
60 |
| Issue:
|
2 |
| Year:
|
2019 |
| Pages:
|
211-218 |
| Summary lang:
|
English |
| . |
| Category:
|
math |
| . |
| Summary:
|
We prove that each linearly continuous function $f$ on $\mathbb R^n$ (i.e., each function continuous on all lines) belongs to the first Baire class, which answers a problem formulated by K.\,C. Ciesielski and D. Miller (2016). The same result holds also for $f$ on an arbitrary Banach space $X$, if $f$ has moreover the Baire property. We also prove (extending a known finite-dimensional result) that such $f$ on a separable $X$ is continuous at all points outside a first category set which is also null in any usual sense. (English) |
| Keyword:
|
linear continuity |
| Keyword:
|
Baire class one |
| Keyword:
|
discontinuity set |
| Keyword:
|
Banach space |
| MSC:
|
26B05 |
| MSC:
|
46B99 |
| idZBL:
|
Zbl 07144890 |
| idMR:
|
MR3982469 |
| DOI:
|
10.14712/1213-7243.2019.003 |
| . |
| Date available:
|
2019-08-05T09:47:43Z |
| Last updated:
|
2021-07-05 |
| Stable URL:
|
http://hdl.handle.net/10338.dmlcz/147816 |
| . |
| Reference:
|
[1] 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:
|
[2] 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:
|
[3] Kuratowski K.: Topology. Vol. I.Academic Press, New York, Państwowe Wydawnictwo Naukowe, Warszawa, 1966. |
| Reference:
|
[4] Lebesgue H.: Sur les fonctions représentable analytiquement.J. Math. Pure Appl. (6) 1 (1905), 139–212 (French). |
| Reference:
|
[5] Lukeš J., Malý J., Zajíček L.: Fine Topology Methods in Real Analysis and Potential Theory.Lecture Notes in Mathematics, 1189, Springer, Berlin, 1986. Zbl 0607.31001, MR 0861411, 10.1007/BFb0075905 |
| Reference:
|
[6] Massera J. L., Schäffer J. J.: Linear differential equations and functional analysis. I.Ann. of Math. (2) 67 (1958), 517–573. MR 0096985, 10.2307/1969871 |
| Reference:
|
[7] Shkarin S. A.: Points of discontinuity of Gateaux-differentiable mappings.Sibirsk. Mat. Zh. 33 (1992), no. 5, 176–185 (Russian); translation in Siberian Math. J. 33 (1992), no. 5, 905–913. MR 1197083 |
| Reference:
|
[8] Slobodnik S. G.: Expanding system of linearly closed sets.Mat. Zametki 19 (1976), 67–84 (Russian); translation in Math. Notes 19 (1976), 39–48. MR 0409742 |
| Reference:
|
[9] Zajíček L.: On the points of multivaluedness of metric projections in separable Banach spaces.Comment. Math. Univ. Carolin. 19 (1978), no. 3, 513–523. MR 0508958 |
| Reference:
|
[10] Zajíček L.: On $\sigma$-porous sets in abstract spaces.Abstr. Appl. Anal. 2005 (2005), no. 5, 509–534. Zbl 1098.28003, MR 2201041, 10.1155/AAA.2005.509 |
| Reference:
|
[11] Zajíček L.: Generic Fréchet differentiability on Asplund spaces via a.e. strict differentiability on many lines.J. Convex Anal. 19 (2012), no. 1, 23–48. MR 2934114 |
| . |
