On Kantorovich's result on the symmetry of Dini derivatives

Martin Koc; Luděk Zajíček

Commentationes Mathematicae Universitatis Carolinae (2010)

  • Volume: 51, Issue: 4, page 619-629
  • ISSN: 0010-2628

Abstract

top
For f : ( a , b ) → 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 σ -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 σ -strongly right porous set A ⊂ ( a , b ) for which no continuous f with A ⊂ 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 ↗ A . This characterization improves both results mentioned above.

How to cite

top

Koc, Martin, and Zajíček, Luděk. "On Kantorovich's result on the symmetry of Dini derivatives." Commentationes Mathematicae Universitatis Carolinae 51.4 (2010): 619-629. <http://eudml.org/doc/246997>.

@article{Koc2010,
abstract = {For $f:(a,b)\rightarrow \mathbb \{R\}$, 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.},
author = {Koc, Martin, Zajíček, Luděk},
journal = {Commentationes Mathematicae Universitatis Carolinae},
keywords = {Dini derivative; one-sided Lipschitzness; $\sigma $-porous set; strong right porosity; abstract porosity; Dini derivative; one-sided Lipschitzness; -porous set; strong right porosity; abstract porosity},
language = {eng},
number = {4},
pages = {619-629},
publisher = {Charles University in Prague, Faculty of Mathematics and Physics},
title = {On Kantorovich's result on the symmetry of Dini derivatives},
url = {http://eudml.org/doc/246997},
volume = {51},
year = {2010},
}

TY - JOUR
AU - Koc, Martin
AU - Zajíček, Luděk
TI - On Kantorovich's result on the symmetry of Dini derivatives
JO - Commentationes Mathematicae Universitatis Carolinae
PY - 2010
PB - Charles University in Prague, Faculty of Mathematics and Physics
VL - 51
IS - 4
SP - 619
EP - 629
AB - For $f:(a,b)\rightarrow \mathbb {R}$, 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.
LA - eng
KW - Dini derivative; one-sided Lipschitzness; $\sigma $-porous set; strong right porosity; abstract porosity; Dini derivative; one-sided Lipschitzness; -porous set; strong right porosity; abstract porosity
UR - http://eudml.org/doc/246997
ER -

References

top
  1. Doležal M., Zelený M., Infinite games and σ -porosity, preprint. 
  2. Kantorovich L.V., Sur les nombres dérivés des fonctions continues, (in Russian), Mat. Sb. 39 (1932), 153–170. 
  3. Oxtoby J.C., Measure and Category, Springer, New York-Berlin, 1980. Zbl0435.28011MR0584443
  4. Thomson B.S., Real Functions, Lecture Notes in Mathematics, 1170, Springer, Berlin, 1985. Zbl0809.26001MR0818744
  5. Zajíček L., On the symmetry of Dini derivates of arbitrary functions, Comment. Math. Univ. Carolin. 22 (1981), 195–209. MR0609947
  6. Zajíček L., Porosity and σ -porosity, Real Anal. Exchange 13 (1987/88), 314–350. MR0943561
  7. Zajíček L., Zelený M., 10.1155/AAA.2005.221, Abstr. Appl. Anal. 2005, 221–227. MR2197116DOI10.1155/AAA.2005.221
  8. Zelený M., Zajíček L., 10.4064/fm185-1-2, Fund. Math. 185 (2005), 19–39. MR2161750DOI10.4064/fm185-1-2

NotesEmbed ?

top

You must be logged in to post comments.

To embed these notes on your page include the following JavaScript code on your page where you want the notes to appear.

Only the controls for the widget will be shown in your chosen language. Notes will be shown in their authored language.

Tells the widget how many notes to show per page. You can cycle through additional notes using the next and previous controls.

    
                

                
Note: Best practice suggests putting the JavaScript code just before the closing </body> tag.