| Title:
|
On Kantorovich's result on the symmetry of Dini derivatives (English) |
| Author:
|
Koc, Martin |
| Author:
|
Zajíček, Luděk |
| Language:
|
English |
| Journal:
|
Commentationes Mathematicae Universitatis Carolinae |
| ISSN:
|
0010-2628 (print) |
| ISSN:
|
1213-7243 (online) |
| Volume:
|
51 |
| Issue:
|
4 |
| Year:
|
2010 |
| Pages:
|
619-629 |
| Summary lang:
|
English |
| . |
| Category:
|
math |
| . |
| Summary:
|
For $f:(a,b)\to \mathbb R$, let $A_f$ be the set of points at which $f$ is Lipschitz from the left but not from the right. L.V. Kantorovich (1932) proved that, if $f$ is continuous, then $A_f$ is a “($k_d$)-reducible set”. The proofs of L. Zajíček (1981) and B.S. Thomson (1985) give that $A_f$ is a $\sigma$-strongly right porous set for an arbitrary $f$. We discuss connections between these two results. The main motivation for the present note was the observation that Kantorovich's result implies the existence of a $\sigma$-strongly right porous set $A\subset (a,b)$ for which no continuous $f$ with $A\subset A_f$ exists. Using Thomson's proof, we prove that such continuous $f$ (resp. an arbitrary $f$) exists if and only if there exist strongly right porous sets $A_n$ such that $A_n\nearrow A$. This characterization improves both results mentioned above. (English) |
| Keyword:
|
Dini derivative |
| Keyword:
|
one-sided Lipschitzness |
| Keyword:
|
$\sigma$-porous set |
| Keyword:
|
strong right porosity |
| Keyword:
|
abstract porosity |
| MSC:
|
26A27 |
| MSC:
|
28A05 |
| idZBL:
|
Zbl 1224.26021 |
| idMR:
|
MR2858265 |
| . |
| Date available:
|
2010-11-30T16:23:19Z |
| Last updated:
|
2013-09-22 |
| Stable URL:
|
http://hdl.handle.net/10338.dmlcz/140842 |
| . |
| Reference:
|
[1] Doležal M., Zelený M.: Infinite games and $\sigma$-porosity.preprint. |
| Reference:
|
[2] Kantorovich L.V.: Sur les nombres dérivés des fonctions continues.(in Russian), Mat. Sb. 39 (1932), 153–170. |
| Reference:
|
[3] Oxtoby J.C.: Measure and Category.Springer, New York-Berlin, 1980. Zbl 0435.28011, MR 0584443 |
| Reference:
|
[4] Thomson B.S.: Real Functions.Lecture Notes in Mathematics, 1170, Springer, Berlin, 1985. Zbl 0809.26001, MR 0818744 |
| Reference:
|
[5] Zajíček L.: On the symmetry of Dini derivates of arbitrary functions.Comment. Math. Univ. Carolin. 22 (1981), 195–209. MR 0609947 |
| Reference:
|
[6] Zajíček L.: Porosity and $\sigma$-porosity.Real Anal. Exchange 13 (1987/88), 314–350. MR 0943561 |
| Reference:
|
[7] Zajíček L., Zelený M.: Inscribing closed non-$\sigma$-lower porous sets into Suslin non-$\sigma$-lower porous sets.Abstr. Appl. Anal. 2005, 221–227. MR 2197116, 10.1155/AAA.2005.221 |
| Reference:
|
[8] Zelený M., Zajíček L.: Inscribing compact non-$\sigma$-porous sets into analytic non-$\sigma$-porous sets.Fund. Math. 185 (2005), 19–39. MR 2161750, 10.4064/fm185-1-2 |
| . |
