# Maslyuchenko V. K.

### Construction of intermediate differentiable functions

Maslyuchenko V. K., Mel'nik V. S.

↓ Abstract

Ukr. Mat. Zh. - 2018. - 70, № 5. - pp. 672-681

For given upper and lower semicontinuous real-valued functions $g$ and $h$, respectively, defined on a closed parallelepiped $X$ in $R^n$ and such that $g(x) < h(x)$ on $X$ and points $x_0 \in X$ and $y_0 \in (g(x_0), h(x_0))$, we construct a smooth function $f : X \rightarrow R$ such that $f(x_0) = y_0$ and $g(x) < f(x) < h(x)$ on $X$. We also present similar constructions for functions defined on separable Hilbert spaces and Asplund spaces.

### Haar’s condition and joint polynomiality of separate polynomial functions

Kosovan V. M., Maslyuchenko V. K., Voloshyn H. A.

Ukr. Mat. Zh. - 2017. - 69, № 1. - pp. 17-27

For systems of functions $F = \{ f_n \in K^X : n \in N\}$ and $G = \{ g_n \in K^Y : n \in N\}$ we consider an $F$ -polynomial $f = \sum^n_{k=1}\lambda_k f_k$, a $G$-polynomial $h = \sum^n_{k,j=1} \lambda_{k,j} f_k \otimes g_j$, and an $F \otimes G$-polynomial $(f_k\otimes g_j)(x, y) = = f_k(x)g_j(y)$, where $(f_k\otimes g_j)(x, y) = f_k(x)g_j(y)$. By using the well-known Haar’s condition from the approximation theory we study the following question: under what assumptions every function $h : X \times Y \rightarrow K$, such that all $x$-sections $h^x = h(x, \cdot )$ are $G$-polynomials and all $y$-sections $h_y = h(\cdot , y)$ are $F$ -polynomials, is an $F \otimes G$-polynomialy. A similar problem is investigated for functions of $n$ variables.

### Sequential closure of the space of jointly continuous functions in the space of separately continuous functions

Maslyuchenko V. K., Voloshyn H. A.

Ukr. Mat. Zh. - 2016. - 68, № 2. - pp. 156-161

Given compact spaces $X$ and $Y$, we study the space $S(X \times Y )$ of separately continuous functions $f : X \times Y \rightarrow R$ endowed with the locally convex topology generated by the seminorms $|| f||^x = \mathrm{max}_{y \in Y} |f(x, y)|,\; x \in X$, and $|| f||_y = \mathrm{max}_{x \in X} |f(x, y)|,\; y \in Y$. Under the assumption that the compact space $X$ is metrizable, we prove that a separately continuous function $f : X \times Y \rightarrow R$ is the limit of a sequence $(f_n)^{\infty}_{n=1}$ of jointly continuous function $f_n : X \times Y \rightarrow R$ in $S(X \times Y )$ provided that the set $D(f)$ of discontinuity points of $f$ has countable projections on $X$.

### Properties of the Ceder Product

Maslyuchenko O. V., Maslyuchenko V. K., Myronyk O. D.

Ukr. Mat. Zh. - 2015. - 67, № 6. - pp. 780-787

We study properties of the Ceder product $X ×_b Y$ of topological spaces $X$ and $Y$, where $b ∈ Y$, recently introduced by the authors. Important examples of the Ceder product are the Ceder plane and the Alexandroff double circle. In particular, for $i = 0, 1, 2, 3$ we establish necessary and sufficient conditions for the Ceder product to be a $T_i$ -space. We prove that the Ceder product $X ×_b Y$ is metrizable if and only if the spaces $X$ and $\overset{.}{Y}=Y\backslash \left\{b\right\}$ are metrizable, $X$ is $σ$-discrete, and the set $\{b\}$ is closed in $Y$. If $X$ is not discrete, then the point $b$ has a countable base of closed neighborhoods in $Y$.

### Points of joint continuity and large oscillations

Maslyuchenko V. K., Nesterenko V. V.

Ukr. Mat. Zh. - 2010. - 62, № 6. - pp. 791–800

For topological spaces $X$ and $Y$ and a metric space $Z$, we introduce a new class $N(X × Y,Z)$ of mappings $f:\; X × Y → Z$ containing all horizontally quasicontinuous mappings continuous with respect to the second variable. It is shown that, for each mapping $f$ from this class and any countable-type set $B$ in $Y$, the set $C_B (f)$ of all points $x$ from $X$ such that $f$ is jointly continuous at any point of the set $\{x\} × B$ is residual in $X$: We also prove that if $X$ is a Baire space, $Y$ is a metrizable compact set, $Z$ is a metric space, and $f ∈ N(X×Y,Z)$, then, for any $ε > 0$, the projection of the set $D^{ε} (f)$ of all points $p ∈ X × Y$ at which the oscillation $ω_f (p) ≥ ε$ onto $X$ is a closed set nowhere dense in $X$.

### Joint continuity of $K_h C$-functions with values in moore spaces

Filipchuk O. I., Maslyuchenko V. K., Mykhailyuk V. V.

Ukr. Mat. Zh. - 2008. - 60, № 11. - pp. 1539 – 1547

We introduce a notion of a categorical cliquish mapping and prove that, for each $K_h C$-mapping $f : X \times Y \rightarrow Z$ (here, $X$ is a topological space, $Y$ is a first countable space, and $Z$ is a Moore space) with categorical cliquish horizontal $y$-sections $f_y$ , the sets $C_y (f)$ are residual $G_\delta$-sets in $X$ for each $y \in Y.$

### Separately continuous mappings with values in nonlocally convex spaces

Karlova O. O., Maslyuchenko V. K.

Ukr. Mat. Zh. - 2007. - 59, № 12. - pp. 1639–1646

We prove that the collection $(X, Y, Z)$ is the Lebesgue triple if $X$ is a metrizable space, $Y$ is a perfectly normal space, and $Z$ is a strongly $\sigma$-metrizable topological vector space with stratification $(Z_m)^{\infty}_{m=1}$, where, for every $m \in \mathbb{N}$, $Z_m$ is a closed metrizable separable subspace of $Z$ arcwise connected and locally arcwise connected.

### Constancy of upper-continuous two-valued mappings into the Sorgenfrey line

Fotii O. H., Maslyuchenko V. K.

Ukr. Mat. Zh. - 2007. - 59, № 8. - pp. 1034–1039

By using the Sierpiński continuum theorem, we prove that every upper-continuous two-valued mapping of a linearly connected space (or even a c-connected space, i.e., a space in which any two points can be connected by a continuum) into the Sorgenfrey line is necessarily constant.

### Direct and Inverse Problems of Baire Classification of Integrals Depending on a Parameter

Banakh T. O., Kutsak S. M., Maslyuchenko O. V., Maslyuchenko V. K.

Ukr. Mat. Zh. - 2004. - 56, № 11. - pp. 1443-1457

We study the problem of the Baire classification of integrals *g* (*y*) = (*If*)(*y*) = ∫ _{X} *f*(*x, y*)*d*μ(*x*), where *y* is a parameter that belongs to a topological space *Y* and *f* are separately continuous functions or functions similar to them. For a given function *g*, we consider the inverse problem of constructing a function *f* such that *g* = *If*. In particular, for compact spaces *X* and *Y* and a finite Borel measure μ on *X*, we prove the following result: In order that there exist a separately continuous function *f* : *X* × *Y* → ℝ such that *g* = *If*, it is necessary and sufficient that all restrictions *g*|_{ Y } _{ n } of the function *g*: *Y* → ℝ be continuous for some closed covering { *Y* _{ n } *: n* ∈ ℕ} of the space *Y*.

### Separately continuous functions with respect to a variable frame

Herasymchuk V. H., Maslyuchenko O. V., Maslyuchenko V. K.

Ukr. Mat. Zh. - 2004. - 56, № 9. - pp. 1281-1286

We show that the set *D*(*f*) of discontinuity points of a function *f* : **R** ^{2} → **R** continuous at every point *p* with respect to two variable linearly independent directions *e* _{1}(*p*) and *e* _{2}(*p*) is a set of the first category. Furthermore, if *f* is differentiable along one of directions, then *D*(*f*) is a nowhere dense set.

### Lebesgue–Cech Dimensionality and Baire Classification of Vector-Valued Separately Continuous Mappings

Kalancha A. K., Maslyuchenko V. K.

Ukr. Mat. Zh. - 2003. - 55, № 11. - pp. 1576-1579

For a metrizable space *X* with finite Lebesgue–Cech dimensionality, a topological space *Y*, and a topological vector space *Z*, we consider mappings *f*: *X* × *Y* → *Z* continuous in the first variable and belonging to the Baire class α in the second variable for all values of the first variable from a certain set everywhere dense in *X*. We prove that every mapping of this type belongs to the Baire class α + 1.

### Joint Continuity and Quasicontinuity of Horizontally Quasicontinuous Mappings

Maslyuchenko V. K., Nesterenko V. V.

Ukr. Mat. Zh. - 2000. - 52, № 12. - pp. 1711-1714

We show that if *X*is a topological space, *Y*satisfies the second axiom of countability, and *Z*is a metrizable space, then, for every mapping *f*: *X*× *Y*→ *Z*that is horizontally quasicontinuous and continuous in the second variable, a set of points *x*∈ *X*such that *f*is continuous at every point from {*x*} × *Y*is residual in *X*. We also generalize a result of Martin concerning the quasicontinuity of separately quasicontinuous mappings.

### New Generalizations of the Scorza-Dragoni Theorem

Gaidukevich O. L., Maslyuchenko V. K.

Ukr. Mat. Zh. - 2000. - 52, № 7. - pp. 881-888

We consider Carathéodory functions *f* : *T* × *X* → *Y*, where *T* is a topological space with regular σ-finite measure, the spaces *X* and *Y* are metrizable and separable, and *X* is locally compact. We show that every function of this sort possesses the Scorza-Dragoni property. A similar result is also established in the case where the space *T* is locally compact and *X* = ℝ^{∞}.

### Characterization of the sets of discontinuity points of separately continuous functions of many variables on the products of metrizable spaces

Maslyuchenko V. K., Mykhailyuk V. V.

Ukr. Mat. Zh. - 2000. - 52, № 6. - pp. 740–747

We show that a subset of the product of*n* metrizable spaces is the set of discontinuity points of some separately continuous function if and only if this subset can be represented in the form of the union of a sequence of*F* _{σ}-sets each, of which is locally projectively a set of the first category.

### Construction of a separately continuous function with given oscillation

Maslyuchenko O. V., Maslyuchenko V. K.

Ukr. Mat. Zh. - 1998. - 50, № 7. - pp. 948–959

We investigate the problem of construction of a separately continuous function *f* whose oscillation is equal to a given nonnegative function *g*. We show that, in the case of a metrizable Baire product, the problem under consideration is solvable if and only if *g* is upper semicontinuous and its support can be covered by countably many sets, which are locally contained in products of sets of the first category.

### A property of partial derivatives

Ukr. Mat. Zh. - 1987. - 39, № 4. - pp. 529–531

### Embeddability conditions for some spaces (a categorical approach)

Ukr. Mat. Zh. - 1984. - 36, № 3. - pp. 316 - 321

### Conditions for the inclusions of intersections and unions of spaces *L*_{p} ( μ ) with a weight

_{p}

Ukr. Mat. Zh. - 1982. - 34, № 4. - pp. 518—522