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On sets where \(\operatorname{lip} f\) is finite. (English) Zbl 1423.26006

Summary: Given a function \(f\colon \mathbb{R}\to \mathbb{R}\), the so-called “little lip” function \(\operatorname{lip} f\) is defined as follows: \[ \operatorname{lip} f(x)=\liminf_{r\searrow 0}\sup_{|x-y|\le r} \frac{|{f(y)-f(x)}|}{r}. \] We show that if \(f\) is continuous on \(\mathbb{R}\), then the set where \(\operatorname{lip} f\) is infinite is a countable union of countable intersections of closed sets (that is, an \(F_{\sigma \delta}\) set). On the other hand, given a countable union \(E\) of closed sets, we construct a continuous function \(f\) such that \(\operatorname{lip} f\) is infinite exactly on \(E\). A further result is that, for a typical continuous function \(f\) on the real line, \(\operatorname{lip} f\) vanishes almost everywhere.

MSC:

26A21 Classification of real functions; Baire classification of sets and functions
26A99 Functions of one variable
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References:

[1] Z. M. Balogh and M. Csörnyei, Scaled-oscillation and regularity, Proc. Amer. Math. Soc. 134 (2006), 2667-2675. · Zbl 1100.26007
[2] S. Banach, Über die Baire’sche Kategorie gewisser Funktionenmengen, Studia Math. 3 (1931), 174-179. · JFM 57.0305.05
[3] Z. Buczolich, Micro tangent sets of continuous functions, Math. Bohem. 128 (2003), 147-167. · Zbl 1027.26003
[4] J. Cheeger, Differentiability of Lipschitz functions on metric measure spaces, Geom. Funct. Anal. 9 (1999), 428-517. · Zbl 0942.58018
[5] B. Hanson, Linear dilatation and differentiability of homeomorphisms of Rn, Proc. Amer. Math. Soc. 140 (2012), 3541-3547. · Zbl 1282.30013
[6] S. Keith, A differentiable structure for metric measure spaces, Adv. Math. 183 (2004), 271-315. · Zbl 1077.46027
[7] W. Stepanoff, Über totale Differenzierbarkeit, Math. Ann. 90 (1923), 318-320. · JFM 49.0183.01
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