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1
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A Lipschitz function which is $C^{∞}$ on a.e. line need not be generically differentiable

100%
Zajíček L.
Colloquium Mathematicum
|
2013
|
tom 131
|
nr 1
29-39
EN
We construct a Lipschitz function f on X = ℝ ² such that, for each 0 ≠ v ∈ X, the function f is $C^{∞}$ smooth on a.e. line parallel to v and f is Gâteaux non-differentiable at all points of X except a first category set. Consequently, the same holds if X (with dimX > 1) is an arbitrary Banach space and "a.e." has any usual "measure sense". This example gives an answer to a natural question concerning the author's recent study of linearly essentially smooth functions (which generalize essentially smooth functions of Borwein and Moors).
2
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Approximate differentiation: Jarník points

64%
Malý J., Zajíček L.
Fundamenta Mathematicae
|
1991-1992
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tom 140
|
nr 1
87-97
EN
We investigate Jarník's points for a real function f defined in ℝ, i.e. points x for which $ap_{y → x}|(f(y)-f(x))/(y-x)|=+∞$. In 1970, Berman has proved that the set $J_f$ of all Jarník's points for a path f of the one-dimensional Brownian motion is the whole ℝ almost surely. We give a simple explicit construction of a continuous function f with $J_f = ℝ. The main result of our paper says that for a typical continuous function f on [0,1] the set $J_f$ is c-dense in [0,1].
3
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Inscribing compact non-σ-porous sets into analytic non-σ-porous sets

64%
Zelený M., Zajíček L.
Fundamenta Mathematicae
|
2005
|
tom 185
|
nr 1
19-39
EN
The main aim of this paper is to give a simpler proof of the following assertion. Let A be an analytic non-σ-porous subset of a locally compact metric space, E. Then there exists a compact non-σ-porous subset of A. Moreover, we prove the above assertion also for σ-P-porous sets, where P is a porosity-like relation on E satisfying some additional conditions. Our result covers σ-⟨g⟩-porous sets, σ-porous sets, and σ-symmetrically porous sets.
4
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A criterion of Γ-nullness and differentiability of convex and quasiconvex functions

64%
Tišer J., Zajíček L.
Studia Mathematica
|
2015
|
tom 227
|
nr 2
149-164
EN
We introduce a criterion for a set to be Γ-null. Using it we give a shorter proof of the result that the set of points where a continuous convex function on a separable Asplund space is not Fréchet differentiable is Γ-null. Our criterion also implies a new result about Gâteaux (and Hadamard) differentiability of quasiconvex functions.
5
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Whitney arcs and 1-critical arcs

51%
Csörnyei M., Kališ J., Zajíček L.
Fundamenta Mathematicae
|
2008
|
tom 199
|
nr 2
119-130
EN
A simple arc γ ⊂ ℝⁿ is called a Whitney arc if there exists a non-constant real function f on γ such that $lim_{y→x, y∈γ} |f(y)-f(x)|/|y-x| = 0$ for every x ∈ γ; γ is 1-critical if there exists an f ∈ C¹(ℝⁿ) such that f'(x) = 0 for every x ∈ γ and f is not constant on γ. We show that the two notions are equivalent if γ is a quasiarc, but for general simple arcs the Whitney property is weaker. Our example also gives an arc γ in ℝ² each of whose subarcs is a monotone Whitney arc, but which is not a strictly monotone Whitney arc. This answers completely a problem of G. Petruska which was solved for n ≥ 3 by the first author in 1999.
6
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A C1 function which is nowhere strongly paraconvex and nowhere semiconcave

38%
Zajíček L.
Control and Cybernetics
|
2007
|
tom 36
|
nr 3
803-810
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