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Keywords:
Dini derivative; one-sided Lipschitzness; $\sigma$-porous set; strong right porosity; abstract porosity
Dini derivative; one-sided Lipschitzness; $\sigma$-porous set; strong right porosity; abstract porosity
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.
References:
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