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A Lipschitz function which is C on a.e. line need not be generically differentiable

Luděk Zajíček — 2013

Colloquium Mathematicae

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...

A note on propagation of singularities of semiconcave functions of two variables

Luděk Zajíček — 2010

Commentationes Mathematicae Universitatis Carolinae

P. Albano and P. Cannarsa proved in 1999 that, under some applicable conditions, singularities of semiconcave functions in n propagate along Lipschitz arcs. Further regularity properties of these arcs were proved by P. Cannarsa and Y. Yu in 2009. We prove that, for n = 2 , these arcs are very regular: they can be found in the form (in a suitable Cartesian coordinate system) ψ ( x ) = ( x , y 1 ( x ) - y 2 ( x ) ) , x [ 0 , α ] , where y 1 , y 2 are convex and Lipschitz on [ 0 , α ] . In other words: singularities propagate along arcs with finite turn.

On differentiability properties of Lipschitz functions on a Banach space with a Lipschitz uniformly Gâteaux differentiable bump function

Luděk Zajíček — 1997

Commentationes Mathematicae Universitatis Carolinae

We improve a theorem of P.G. Georgiev and N.P. Zlateva on Gâteaux differentiability of Lipschitz functions in a Banach space which admits a Lipschitz uniformly Gâteaux differentiable bump function. In particular, our result implies the following theorem: If d is a distance function determined by a closed subset A of a Banach space X with a uniformly Gâteaux differentiable norm, then the set of points of X A at which d is not Gâteaux differentiable is not only a first category set, but it is even σ -porous...

Singular points of order k of Clarke regular and arbitrary functions

Luděk Zajíček — 2012

Commentationes Mathematicae Universitatis Carolinae

Let X be a separable Banach space and f a locally Lipschitz real function on X . For k , let Σ k ( f ) be the set of points x X , at which the Clarke subdifferential C f ( x ) is at least k -dimensional. It is well-known that if f is convex or semiconvex (semiconcave), then Σ k ( f ) can be covered by countably many Lipschitz surfaces of codimension k . We show that this result holds even for each Clarke regular function (and so also for each approximately convex function). Motivated by a resent result of A.D. Ioffe, we prove...

Remarks on Fréchet differentiability of pointwise Lipschitz, cone-monotone and quasiconvex functions

Luděk Zajíček — 2014

Commentationes Mathematicae Universitatis Carolinae

We present some consequences of a deep result of J. Lindenstrauss and D. Preiss on Γ -almost everywhere Fréchet differentiability of Lipschitz functions on c 0 (and similar Banach spaces). For example, in these spaces, every continuous real function is Fréchet differentiable at Γ -almost every x at which it is Gâteaux differentiable. Another interesting consequences say that both cone-monotone functions and continuous quasiconvex functions on these spaces are Γ -almost everywhere Fréchet differentiable....

A remark on functions continuous on all lines

Luděk Zajíček — 2019

Commentationes Mathematicae Universitatis Carolinae

We prove that each linearly continuous function f on n (i.e., each function continuous on all lines) belongs to the first Baire class, which answers a problem formulated by K. C. Ciesielski and D. Miller (2016). The same result holds also for f on an arbitrary Banach space X , if f has moreover the Baire property. We also prove (extending a known finite-dimensional result) that such f on a separable X is continuous at all points outside a first category set which is also null in any usual sense.

A note on intermediate differentiability of Lipschitz functions

Luděk Zajíček — 1999

Commentationes Mathematicae Universitatis Carolinae

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 σ -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.

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