# Maslyuchenko V. K.

### Discontinuity points of separately continuous mappings with at most countable set of values

Filipchuk O. I., Maslyuchenko V. K.

↓ Abstract

Ukr. Mat. Zh. - 2019. - 71, № 6. - pp. 801-807

UDC 517.51

We obtain a general result on the constancy of separately continuous mappings and their analogs, which implies the wellknown
Sierpi´nski theorem. By using this result, we study the set of continuity points of separately continuous mappings
with at most countably many values including, in particular, the mappings defined on the square of the Sorgenfrey line
with values in the Bing plane.

### 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.

### Separately continuous functions on products of compact sets and their dependence on $\mathfrak{n}$ variables

Maslyuchenko V. K., Mykhailyuk V. V.

Ukr. Mat. Zh. - 1995. - 47, № 3. - pp. 344-350

By using the theorem on the density of the topological product and the generalized theorem on the dependence of a continuous function defined on a product of spaces on countably many coordinates, we show that every separately continuous function defined on a product of two spaces representable as products of compact spaces with density $≤ \mathfrak{n}$ depends on n variables. In the case of metrizable compact sets, we obtain a complete description of the sets of discontinuity points for functions of this sort.

### Inverse problems of the theory of separately continuous mappings

Maslyuchenko V. K., Mykhailyuk V. V., Sobchuk V. S.

Ukr. Mat. Zh. - 1992. - 44, № 9. - pp. 1209–1220

The present paper investigates the problem of constructing a separately continuous function defined on the product of two topological spaces that possesses a specified set of points of discontinuity and the related special problem of constructing a pointwise convergent sequence of continuous functions that possesses a specified set of points of nonuniform convergence and set of points of discontinuity of a limit function. In the metrizable case the former problem is solved for separable $F_σ$-sets whose projections onto every cofactor is of the first category. The second problem is solved for a pair of embedded $F_σ$.

### 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