Martin Koc, Luděk Zajíček
On Kantorovich's result on the symmetry of Dini derivatives

Comment.Math.Univ.Carolin. 51,4 (2010) 619-629.

Abstract: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.~Zaj\'\i\v cek (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.

Keywords: Dini derivative, one-sided Lipschitzness, $\sigma$-porous set, strong right porosity, abstract porosity
AMS Subject Classification: 26A27 28A05

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