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Dini derivative; one-sided Lipschitzness; $\sigma$-porous set; strong right porosity; abstract porosity
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.
[1] Doležal M., Zelený M.: Infinite games and $\sigma$-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. MR 0584443 | Zbl 0435.28011
[4] Thomson B.S.: Real Functions. Lecture Notes in Mathematics, 1170, Springer, Berlin, 1985. MR 0818744 | Zbl 0809.26001
[5] Zajíček L.: On the symmetry of Dini derivates of arbitrary functions. Comment. Math. Univ. Carolin. 22 (1981), 195–209. MR 0609947
[6] Zajíček L.: Porosity and $\sigma$-porosity. Real Anal. Exchange 13 (1987/88), 314–350. MR 0943561
[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. DOI 10.1155/AAA.2005.221 | MR 2197116
[8] Zelený M., Zajíček L.: Inscribing compact non-$\sigma$-porous sets into analytic non-$\sigma$-porous sets. Fund. Math. 185 (2005), 19–39. DOI 10.4064/fm185-1-2 | MR 2161750
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