On the structure of universal differentiability sets

Michael Dymond

Commentationes Mathematicae Universitatis Carolinae (2017)

  • Volume: 58, Issue: 3, page 315-326
  • ISSN: 0010-2628

Abstract

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A subset of ℝ d is called a universal differentiability set if it contains a point of differentiability of every Lipschitz function f : ℝ d → ℝ . We show that any universal differentiability set contains a ‘kernel’ in which the points of differentiability of each Lipschitz function are dense. We further prove that no universal differentiability set may be decomposed as a countable union of relatively closed, non-universal differentiability sets.

How to cite

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Dymond, Michael. "On the structure of universal differentiability sets." Commentationes Mathematicae Universitatis Carolinae 58.3 (2017): 315-326. <http://eudml.org/doc/294079>.

@article{Dymond2017,
abstract = {A subset of $\mathbb \{R\}^\{d\}$ is called a universal differentiability set if it contains a point of differentiability of every Lipschitz function $f\colon \mathbb \{R\}^\{d\}\rightarrow \mathbb \{R\}$. We show that any universal differentiability set contains a ‘kernel’ in which the points of differentiability of each Lipschitz function are dense. We further prove that no universal differentiability set may be decomposed as a countable union of relatively closed, non-universal differentiability sets.},
author = {Dymond, Michael},
journal = {Commentationes Mathematicae Universitatis Carolinae},
keywords = {differentiability; Lipschitz functions; universal differentiability set; $\sigma $-porous set},
language = {eng},
number = {3},
pages = {315-326},
publisher = {Charles University in Prague, Faculty of Mathematics and Physics},
title = {On the structure of universal differentiability sets},
url = {http://eudml.org/doc/294079},
volume = {58},
year = {2017},
}

TY - JOUR
AU - Dymond, Michael
TI - On the structure of universal differentiability sets
JO - Commentationes Mathematicae Universitatis Carolinae
PY - 2017
PB - Charles University in Prague, Faculty of Mathematics and Physics
VL - 58
IS - 3
SP - 315
EP - 326
AB - A subset of $\mathbb {R}^{d}$ is called a universal differentiability set if it contains a point of differentiability of every Lipschitz function $f\colon \mathbb {R}^{d}\rightarrow \mathbb {R}$. We show that any universal differentiability set contains a ‘kernel’ in which the points of differentiability of each Lipschitz function are dense. We further prove that no universal differentiability set may be decomposed as a countable union of relatively closed, non-universal differentiability sets.
LA - eng
KW - differentiability; Lipschitz functions; universal differentiability set; $\sigma $-porous set
UR - http://eudml.org/doc/294079
ER -

References

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