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# Article

 Title: A note on intermediate differentiability of Lipschitz functions (English) Author: Zajíček, Luděk Language: English Journal: Commentationes Mathematicae Universitatis Carolinae ISSN: 0010-2628 (print) ISSN: 1213-7243 (online) Volume: 40 Issue: 4 Year: 1999 Pages: 795-799 . Category: math . Summary: Let $f$ be a Lipschitz function on a superreflexive Banach space $X$. We prove that then the set of points of $X$ at which $f$ has no intermediate derivative is not only a first category set (which was proved by M. Fabian and D. Preiss for much more general spaces $X$), but it is even $\sigma$-porous in a rather strong sense. In fact, we prove the result even for a stronger notion of uniform intermediate derivative which was defined by J.R. Giles and S. Sciffer. (English) Keyword: Lipschitz function Keyword: intermediate derivative Keyword: $\sigma$-porous set Keyword: superreflexive Banach space MSC: 46G05 MSC: 58C20 idZBL: Zbl 1010.46042 idMR: MR1756555 . Date available: 2009-01-08T18:57:47Z Last updated: 2012-04-30 Stable URL: http://hdl.handle.net/10338.dmlcz/119133 . Reference: [1] Aronszajn N.: Differentiability of Lipschitzian mappings between Banach spaces.Studia Math. 57 (1976), 147-190. Zbl 0342.46034, MR 0425608 Reference: [2] Bates S.M., Johnson W.B., Lindenstrauss J., Preiss D., Schechtman G.: Affine approximation of Lipschitz functions and non linear quotiens.to appear. Reference: [3] Fabian M., Preiss D.: On intermediate differentiability of Lipschitz functions on certain Banach spaces.Proc. Amer. Math. Soc. 113 (1991), 733-740. Zbl 0743.46040, MR 1074753 Reference: [4] Giles J.R., Sciffer S.: Generalising generic differentiability properties from convex to locally Lipschitz functions.J. Math. Anal. Appl. 188 (1994), 833-854. Zbl 0897.46025, MR 1305489 Reference: [5] Preiss D.: Differentiability of Lipschitz functions in Banach spaces.J. Funct. Anal. 91 (1990), 312-345. MR 1058975 Reference: [6] Preiss D., Zajíček L.: Sigma-porous sets in products of metric spaces and sigma-directionally porous sets in Banach spaces.Real Analysis Exchange 24 (1998-99), 295-313. MR 1691753 Reference: [7] Preiss D., Zajíček L.: Directional derivatives of Lipschitz functions.to appear. MR 1853802 Reference: [8] Zajíček L.: Porosity and $\sigma$-porosity.Real Analysis Exchange 13 (1987-88), 314-350. MR 0943561 Reference: [9] Zajíček L.: On differentiability properties of Lipschitz functions on a Banach space with a Lipschitz uniformly Gâteaux differentiable bump function.Comment. Math. Univ. Carolinae 32 (1997), 329-336. MR 1455499 Reference: [10] Zajíček L.: Small non-sigma-porous sets in topologically complete metric spaces.Colloq. Math. 77 (1998), 293-304. MR 1628994 Reference: [11] Zelený M.: The Banach-Mazur game and $\sigma$-porosity.Fund. Math. 150 (1996), 197-210. MR 1405042 .

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