Abstract
We denote the local “little” and “big” Lipschitz functions of a function f : ℝ → ℝ by lip f and Lip f. In this paper we continue our research concerning the following question. Given a set E ⊂ ℝ is it possible to find a continuous function f such that lip f = 1E or Lip f = 1E?
In giving some partial answers to this question uniform density type (UDT) and strong uniform density type (SUDT) sets play an important role.
In this paper we show that modulo sets of zero Lebesgue measure any measurable set coincides with a Lip 1 set.
On the other hand, we prove that there exists a measurable SUDT set E such that for any Gδ set E͠ satisfying ∣EΔE͠∣ = 0 the set E͠ does not have UDT. Combining these two results we obtain that there exist Lip 1 sets not having UDT, that is, the converse of one of our earlier results does not hold.
Communicated by Tomasz Natkaniec
Acknowledgement
We thank the referees for several comments which improved our paper.
References
[1] Balogh, Z. M.—Csörnyei, M.: Scaled-oscillation and regularity, Proc. Amer. Math. Soc. 134 (2006), 2667–2675.10.1090/S0002-9939-06-08290-6Search in Google Scholar
[2] Buczolich, Z.: Category of density points of Cantor sets, Real Anal. Exchange 29(1) (2003/2004), 497–502.10.14321/realanalexch.29.1.0497Search in Google Scholar
[3] Buczolich, Z.—Hanson, B.—Maga, B.—Vértesy, G.: Big and little Lipschitz one sets, submitted, arxiv: 1905.11081.Search in Google Scholar
[4] Buczolich, Z.—Hanson, B.—Rmoutil, M.—Zürcher, T.: On Sets where lipfis finite, Studia Math. 249 (2019), 33–58.10.4064/sm170820-26-5Search in Google Scholar
[5] Hanson, B.: Linear dilatation and differentiability of homeomorphisms of ℝn, Proc. Amer. Math. Soc. 140 (2012), 3541–3547.10.1090/S0002-9939-2012-11688-0Search in Google Scholar
[6] Hanson, B.: Sets of Non-differentiability for Functions with Finite Lower Scaled Oscillation, Real Anal. Exchange 41(1) (2016), 87–100.10.14321/realanalexch.41.1.0087Search in Google Scholar
[7] Kuratowski, K.: Topologie. Vol. 1, 4th ed., PWN, Warsaw, 1958; English transl., Academic Press, New York; PWN, Warsaw, 1966.Search in Google Scholar
[8] Malý, J.—Zajíček, L.: On Stepanov type differentiability theorems, Acta Math. Hungar. 145(1) (2015), 174–190.10.1007/s10474-014-0465-6Search in Google Scholar
[9] Malý, J.—Zindulka, O.: Mapping analytic sets onto cubes by little Lipschitz functions, Eur. J. Math. 5(1) (2019), 91–105.10.1007/s40879-018-0288-zSearch in Google Scholar
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