On the convergence of -primitives
On the density maxima of a function
On the Differentiability Points of a Function of Two Real Variables Admitting Partial Derivatives.
On the differentiation of integrals of functions from Lφ(L)
On the differentiation of integrals of functions from Orlicz classes
On the linear Denjoy property of two-variable continuous functions
The classical Denjoy-Young-Saks theorem gives a relation, here termed the Denjoy property, between the Dini derivatives of an arbitrary one-variable function that holds almost everywhere. Concerning the possible generalizations to higher dimensions, A. S. Besicovitch proved the following: there exists a continuous function of two variables such that at each point of a set of positive measure there exist continuum many directions, in each of which one Dini derivative is infinite and the other...
On the points of non-differentiability of convex functions
We characterize sets of non-differentiability points of convex functions on . This completes (in ) the result by Zajíček [2] which gives a characterization of the magnitude of these sets.
On the range of the derivative of a real-valued function with bounded support
We study the set f’(X) = f’(x): x ∈ X when f:X → ℝ is a differentiable bump. We first prove that for any C²-smooth bump f: ℝ² → ℝ the range of the derivative of f must be the closure of its interior. Next we show that if X is an infinite-dimensional separable Banach space with a -smooth bump b:X → ℝ such that is finite, then any connected open subset of X* containing 0 is the range of the derivative of a -smooth bump. We also study the finite-dimensional case which is quite different. Finally,...
On the theorems of Y. Mibu and G. Debs on separate continuity.
On weakly Gibson -measurable mappings
A function f: X → Y between topological spaces is said to be a weakly Gibson function if for any open connected set U ⊆ X. We prove that if X is a locally connected hereditarily Baire space and Y is a T₁-space then an -measurable mapping f: X → Y is weakly Gibson if and only if for any connected set C ⊆ X with dense connected interior the image f(C) is connected. Moreover, we show that each weakly Gibson -measurable mapping f: ℝⁿ → Y, where Y is a T₁-space, has a connected graph.
Optimal embeddings of critical Sobolev-Lorentz-Zygmund spaces
We establish the embedding of the critical Sobolev-Lorentz-Zygmund space into the generalized Morrey space with an optimal Young function Φ. As an application, we obtain the almost Lipschitz continuity for functions in . O’Neil’s inequality and its reverse play an essential role in the proofs of the main theorems.
Oscillations, quasi-oscillations and joint continuity.
Parametric extension of the Poincaré theorem
Partial Differentiation, Differentiation and Continuity on n -Dimensional Real Normed Linear Spaces
In this article, we aim to prove the characterization of differentiation by means of partial differentiation for vector-valued functions on n-dimensional real normed linear spaces (refer to [15] and [16]).
Partial Differentiation of Vector-Valued Functions on n -Dimensional Real Normed Linear Spaces
In this article, we define and develop partial differentiation of vector-valued functions on n-dimensional real normed linear spaces (refer to [19] and [20]).
Pettis integration
Polynomial approximation and twenty years later.
Properties of -smooth mappings with one-dimensional gradient range.
Proprietà fini delle funzioni con deformazione limitata ed applicazioni a problemi variazionali
Proto-differentiability of set-valued mappings and its applications in optimization