The C k Space
In this article, we formalize continuous differentiability of realvalued functions on n-dimensional real normed linear spaces. Next, we give a definition of the Ck space according to [23].
The Denjoy-Clarkson property with respect to Hausdorff measures for the gradient mapping of functions of several variables
We construct a differentiable function () such that the set is a nonempty set of Hausdorff dimension . This answers a question posed by Z. Buczolich.
The Differentiable Functions from R into R n
In control engineering, differentiable partial functions from R into Rn play a very important role. In this article, we formalized basic properties of such functions.
The hardy-littlewood maximal function and derivation of integrals.
The local structure of the solutions of the multidimensional translation equation.
The Lusin Theorem and Horizontal Graphs in the Heisenberg Group
In this paper we prove that every collection of measurable functions fα , |α| = m, coincides a.e. withmth order derivatives of a function g ∈ Cm−1 whose derivatives of order m − 1 may have any modulus of continuity weaker than that of a Lipschitz function. This is a stronger version of earlier results of Lusin, Moonens-Pfeffer and Francos. As an application we construct surfaces in the Heisenberg group with tangent spaces being horizontal a.e.
The quasi-real extension of the real numbers
Two classes of Darboux-like, Baire one functions of two variables
Among the many characterizations of the class of Baire one, Darboux real-valued functions of one real variable, the 1907 characterization of Young and the 1997 characterization of Agronsky, Ceder, and Pearson are particularly intriguing in that they yield interesting classes of functions when interpreted in the two-variable setting. We examine the relationship between these two subclasses of the real-valued Baire one defined on the unit square.