We prove the result on Baire classification of mappings $f:X\times Y\to Z$ which are continuous with respect to the first variable and belongs to a Baire class with respect to the second one, where $X$ is a $PP$-space, $Y$ is a topological space and $Z$ is a strongly $\sigma$-metrizable space with additional properties. We show that for any topological space $X$, special equiconnected space $Z$ and a mapping $g:X\to Z$ of the $(n-1)$-th Baire class there exists a strongly separately continuous mapping $f:X^n\to Z$ with the diagonal $g$. For wide classes of spaces...

It is proved that real functions on $ℝ$ which can be represented as the difference of two semiconvex functions with a general modulus (or of two lower $C^1$-functions, or of two strongly paraconvex functions) coincide with semismooth functions on $ℝ$ (i.e. those locally Lipschitz functions on $ℝ$ for which $f_{+}^{\text{'}}\left(x\right)={lim}_{t\to x+}{f}_{+}^{\text{'}}\left(t\right)$ and $f_{-}^{\text{'}}\left(x\right)={lim}_{t\to x-}{f}_{-}^{\text{'}}\left(t\right)$ for each $x$). Further, for each modulus $\omega$, we characterize the class $DS{C}_{\omega }$ of functions on $ℝ$ which can be written as $f=g-h$, where $g$ and $h$ are semiconvex with modulus $C\omega$ (for some $C>0$) using a new notion of...

In this article, we formalize the differentiability of functions from the set of real numbers into a normed vector space [14].

In this article, we formalize differentiability of functions on normed linear spaces. Partial derivative, mean value theorem for vector-valued functions, continuous differentiability, etc. are formalized. As it is well known, there is no exact analog of the mean value theorem for vector-valued functions. However a certain type of generalization of the mean value theorem for vector-valued functions is obtained as follows: If ||ƒ'(x + t · h)|| is bounded for t between 0 and 1 by some constant M, then...

In this article, we define and develop differentiation of vector-valued functions on n-dimensional real normed linear spaces (refer to [16] and [17]).

We give characterizations of the distributional derivatives $D^{1,1}$, $D^{1,0}$, $D^{0,1}$ of functions of two variables of locally finite variation. Then we use these results to prove the existence theorem for the hyperbolic equation with a nonhomogeneous term containing the distributional derivative determined by an additive function of an interval of finite variation. An application of the above theorem to a hyperbolic equation with an impulse effect is also given.

In Example 1, we describe a subset X of the plane and a function on X which has a ${}^k$-extension to the whole ${ℝ}^2$ for each finite, but has no ${}^{\infty }$-extension to ${ℝ}^2$. In Example 2, we construct a similar example of a subanalytic subset of ${ℝ}^5$; much more sophisticated than the first one. The dimensions given here are smallest possible.

It is shown that $n$ times Peano differentiable functions defined on a closed subset of ${ℝ}^m$ and satisfying a certain condition on that set can be extended to $n$ times Peano differentiable functions defined on ${ℝ}^m$ if and only if the $n$th order Peano derivatives are Baire class one functions.

We investigate several extension properties of Fréchet differentiable functions defined on closed sets for o-minimal expansions of real closed fields.

On étudie les fonctions de deux variables réelles qui sont séparément analytiques sur un ouvert du plan. On montre que ces fonctions sont analytiques en tout point du domaine de définition hors d’un fermé de ce domaine dont les projections sur chacun des deux axes de coordonnées sont des ensembles polaires. Inversempent, pour tout tel fermé $F$, on construit une fonction séparément analytique dont le domaine d’analyticité est le complémentaire de $F$.

We introduce a notion of a function of finite fractional variation and characterize such functions together with their weak $\sigma$-additive fractional derivatives. Next, we use these functions to study differential equations of fractional order, containing a $\sigma$-additive term—we prove existence and uniqueness of a solution as well as derive a Cauchy formula for the solution. We apply these results to impulsive equations, i.e. equations containing the Dirac measures.

Some differentiability properties of functions of several variables of finite variation are investigated.