On sets of non-differentiability of Lipschitz and convex functions

Zaj́ıček, Luděk

doi:10.21136/mb.2007.133997

Cited by 5 publications

(5 citation statements)

References 17 publications

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“…Using the same reasoning as in [10,Corollary 11], can obtain the following corollary. It is a consequence of a recent result of Zajíček [19] who proved that the sets in C (in a separable Banach space) are Γ-null (in the sense of [14]) and the resutls of [14].…”

Section: Differentiability Of Cone Monotone Mappingsmentioning

confidence: 93%

Cone monotone mappings: Continuity and differentiability

Duda

2008

Nonlinear Analysis: Theory, Methods & Applications

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We generalize some results of Borwein, Burke, Lewis, and Wang to mappings with values in metric (resp. ordered normed linear) spaces. We define two classes of monotone mappings between an ordered linear space and a metric space (resp. ordered linear space): K-monotone dominated and coneto-cone monotone mappings. K-monotone dominated mappings naturally generalize mappings with finite variation (in the classical sense) and K-monotone functions defined by Borwein, Burke and Lewis, to mappings with domains and ranges of higher dimensions. First, using results of Veselý and Zajíček, we show some relationships between these classes. Then, we show that every K-monotone function f : X → R, where X is any Banach space, is continuous outside of a set which can be covered by countably many Lipschitz hypersurfaces. This sharpens a result due to Borwein and Wang. As a consequence, we obtain a similar result for K-monotone dominated and cone-to-cone monotone mappings. Finally, we prove several results concerning almost everywhere differentiability (also in metric and w * -senses) of these mappings.

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Section: Differentiability Of Cone Monotone Mappingsmentioning

confidence: 93%

Cone monotone mappings: Continuity and differentiability

Duda

2008

Nonlinear Analysis: Theory, Methods & Applications

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“…Such r k exist since ϕ(r) = x, ϕ(s) → ∞ as s → ∞, and ϕ(u) ∈ (x + K β ′ ,x * ) by the choice of ε. Then (12) implies that ϕ(r k ) ≥ K α,x * x + t k v/2, and thus f (ϕ(r k )) ≥ f (x + t k v/2). Now, since ψ is ε-Lipschitz, we have (1 − ε)|r − r k | ≤ ϕ(r k ) − ϕ(r) = t k , and thus…”

Section: Cone Monotone Functionsmentioning

confidence: 96%

“…By [7,Corollary 3.11] there exists a Γ-null B ⊂ X such that g is Fréchet differentiable at each x ∈ X \ B. Since A is Γ-null by [12,Theorem 2.4], we have that A ∪ B is Γ-null and thus there exists x ∈ X \ (A ∪ B).…”

Section: Cone Monotone Functionsmentioning

confidence: 99%

“…By results of [8], Ã is a strict subclass of Aronszajn null sets. Recently, Zajíček [12] proved that the sets in Ã (and even C) are Γ-null, which is a notion of null sets due to Lindenstrauss and Preiss [7] (here, a definition and basic properties of this notion can be found). Thus, Theorem 10 has the following corollary: if X is a Banach space with separable dual (i.e.…”

Section: Introductionmentioning

confidence: 99%

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On Gâteaux Differentiability of Pointwise Lipschitz Mappings

Duda¹

2008

Can. math. bull.

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Abstract. We prove that for every function f : X → Y , where X is a separable Banach space and Y is a Banach space with RNP, there exists a set A ∈Ãsuch that f is Gâteaux differentiable at all x ∈ S( f )\A, where S( f ) is the set of points where f is pointwise-Lipschitz. This improves a result of Bongiorno. As a corollary, we obtain that every K-monotone function on a separable Banach space is Hadamard differentiable outside of a set belonging toC; this improves a result due to Borwein and Wang. Another corollary is that if X is Asplund, f : X → R cone monotone, g : X → R continuous convex, then there exists a point in X, where f is Hadamard differentiable and g is Fréchet differentiable.

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“…(For definition of Aronszajn and Haar null sets see [2].) Moreover, each member of C is also Γ-null (see [20]). The important σ-ideal of Γ-null subsets of X was introduced in [8].…”

Section: The Usual Modern Definition Of the Hadamard Derivative Is Th...mentioning

confidence: 99%

Properties of Hadamard directional derivatives: Denjoy-Young-Saks theorem for functions on Banach spaces

Zajicek

2013

Preprint

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The classical Denjoy-Young-Saks theorem on Dini derivatives of arbitrary functions f : R → R was extended by U.S. Haslam-Jones (1932) andA.J. Ward (1935) to arbitrary functions on R 2 . This extension gives the strongest relation among upper and lower Hadamard directionalwhich holds almost everywhere for an arbitrary function f : R 2 → R. Our main result extends the theorem of Haslam-Jones and Ward to functions on separable Banach spaces.

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On sets of non-differentiability of Lipschitz and convex functions

Abstract: Institute of Mathematics of the Czech Academy of Sciences provides access to digitized documents strictly for personal use. Each copy of any part of this document must contain these Terms of use.

Cited by 5 publications

References 17 publications

Cone monotone mappings: Continuity and differentiability

Cone monotone mappings: Continuity and differentiability

On Gâteaux Differentiability of Pointwise Lipschitz Mappings

Properties of Hadamard directional derivatives: Denjoy-Young-Saks theorem for functions on Banach spaces

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