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**Lipschitz functions with unexpectedly large sets of nondifferentiability points.**
*(English)*
Zbl 1098.26010

It is well known by the Rademacher theorem that a Lipschitz function \(f:\mathbb{R}^n\to\mathbb{R}^m\) is differentiable almost everywhere or conversely: the set of points where \(f\) is not differentiable is very small. For one-dimensional functions, this result is sharp in that sense that for every (Lebesgue) null set \(E\subset\mathbb{R}\) there exists a Lipschitz function which is nondifferentiable on \(E\). This sharpness can not be extended for higher-dimensional spaces. It can be shown that there exists a null set \(E\subset\mathbb{R}^2\) such that every real-valued Lipschitz function on \(\mathbb{R}^2\) is differentiable at some point of \(E\).

The authors provide a special \(G_\delta\) subset \(E\subset \mathbb{R}^2\) containing a dense set of lines for which the set of points of differentiability of particular Lipschitz functions inside of \(E\) is extremely small. More exactly, they construct

(1) two real-valued Lipschitz functions on \(\mathbb{R}^2\) with no common points of differentiability in \(E\) (or equivalently: a vector-valued Lipschitz function on \(\mathbb{R}^2\) which is nondifferentiable on \(E)\),

(2) a real-valued Lipschitz function on \(\mathbb{R}^2\) whose set of points of differentiability in \(E\) is uniformly purely unrectifiable (i.e., this set meets every rectifiable curve in a set of one-dimensional measure zero).

The authors provide a special \(G_\delta\) subset \(E\subset \mathbb{R}^2\) containing a dense set of lines for which the set of points of differentiability of particular Lipschitz functions inside of \(E\) is extremely small. More exactly, they construct

(1) two real-valued Lipschitz functions on \(\mathbb{R}^2\) with no common points of differentiability in \(E\) (or equivalently: a vector-valued Lipschitz function on \(\mathbb{R}^2\) which is nondifferentiable on \(E)\),

(2) a real-valued Lipschitz function on \(\mathbb{R}^2\) whose set of points of differentiability in \(E\) is uniformly purely unrectifiable (i.e., this set meets every rectifiable curve in a set of one-dimensional measure zero).

Reviewer: Jörg Thierfelder (Ilmenau)

### MSC:

26B35 | Special properties of functions of several variables, Hölder conditions, etc. |

26A27 | Nondifferentiability (nondifferentiable functions, points of nondifferentiability), discontinuous derivatives |

49J50 | Fréchet and Gateaux differentiability in optimization |

26B05 | Continuity and differentiation questions |

46G05 | Derivatives of functions in infinite-dimensional spaces |

26A21 | Classification of real functions; Baire classification of sets and functions |

54C50 | Topology of special sets defined by functions |