Geodesic
On Kantorovich's result on the symmetry of Dini derivatives
Commentationes Mathematicae Universitatis Carolinae, Tome 51 (2010) no. 4, pp. 619-629

Voir la notice de l'article provenant de la source Czech Digital Mathematics Library

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.
Classification : 26A27, 28A05
Keywords: Dini derivative; one-sided Lipschitzness; $\sigma$-porous set; strong right porosity; abstract porosity 
@article{CMUC_2010__51_4_a6,
     author = {Koc, Martin and Zaj{\'\i}\v{c}ek, Lud\v{e}k},
     title = {On {Kantorovich's} result on the symmetry of {Dini} derivatives},
     journal = {Commentationes Mathematicae Universitatis Carolinae},
     pages = {619--629},
     publisher = {mathdoc},
     volume = {51},
     number = {4},
     year = {2010},
     mrnumber = {2858265},
     zbl = {1224.26021},
     language = {en},
     url = {http://geodesic.mathdoc.fr/item/CMUC_2010__51_4_a6/}
}
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
SP  - 619
EP  - 629
VL  - 51
IS  - 4
PB  - mathdoc
UR  - http://geodesic.mathdoc.fr/item/CMUC_2010__51_4_a6/
LA  - en
ID  - CMUC_2010__51_4_a6
ER  - 
%0 Journal Article
%A Koc, Martin
%A Zajíček, Luděk
%T On Kantorovich's result on the symmetry of Dini derivatives
%J Commentationes Mathematicae Universitatis Carolinae
%D 2010
%P 619-629
%V 51
%N 4
%I mathdoc
%U http://geodesic.mathdoc.fr/item/CMUC_2010__51_4_a6/
%G en
%F CMUC_2010__51_4_a6
Koc, Martin; Zajíček, Luděk. On Kantorovich's result on the symmetry of Dini derivatives. Commentationes Mathematicae Universitatis Carolinae, Tome 51 (2010) no. 4, pp. 619-629. http://geodesic.mathdoc.fr/item/CMUC_2010__51_4_a6/