The cluster sets of continuous functions

O. V. Maslyuchenko; D. P. Onypa

doi:10.15330/ms.46.1.44-50

Let $X$ be a metrizable space, $D$ be a boundary locally connected open subset of $X$ (i.e. for any $x\in \overline{D}\setminus D$ and for any neighborhood $U$ of $x$ there is a neighborhood $V$ of $x$ such that $V\subseteq U$ and $V\cap D$ is connected), $L$ be a closed subset of $\overline{D}\setminus D$ and $\Phi\colon L\multimap \overline{\mathbb R}$ be a multivalued mapping. We proved that there is a continuous function $f\colon D\to \mathbb R$ such that the cluster set $\overline{f}(x)=\Phi(x)$ for any $x\in L$ if and only if $\Phi$ is upper continuous compact-valued multifunction and $\Phi(x)$ is connected for any $x\in L$.

1. P. Painleve, Lecons sur la theorie analytique des equations differentielles professees a Stockholm, Paris, 1897.

2. P. Kostyrko, Some properties of oscillation, Math. Slovaca, 30 (1980), 157–162.

3. Z. Duszhynski, Z. Grande, S. Ponomarev, On the $\omega$-primitives, Math. Slovaka, 51 (2001), 469–476.

4. J. Ewert, S. Ponomarev, On the exsistance of $\omega$-primitives on arbitrary metric spaces, Math. Slovaca, 53 (2003), 51–57.

5. J. Ewert, S. Ponomarev, Oscillation and $\omega$-primiteves, Real Anal. Exchange, 26 (2001–2002), 687–702.

6. S. Kowalczyk, The $\omega$-problem, Diss. math. (Rozpr. mat.), 501 (2014), 1–55.

7. J. Borsik, On quasioscillation, Tatra Mountains Math. Publ., 2 (1993), 25–36.

8. O.V. Maslyuchenko, V.K. Maslyuchenko, The construction of a separately continuous function with given oscillation, Ukr. Math. J., 50 (1998), №7, 948–959. (in Ukrainian)

9. O.V. Maslyuchenko, The oscillation of quasi-continuous functions on pairwise attainable spaces, Houston Journal of Mathematics, 35 (2009), №1, 113–130.

10. O.V. Maslyuchenko, The construction of $\omega$-primitives: the oscillation of sum of functions, Mat. Visn. Nauk. Tov. Im. Shevchenka, 5 (2008), 151–163. (in Ukrainian)

11. O.V. Maslyuchenko The construction of $\omega$-primitives: strongly attainable spaces, Mat. Visn. Nauk. Tov. Im. Shevchenka, 6 (2009), 155–178. (in Ukrainian)

12. O.V. Maslyuchenko, Fluctuations of almost continuous functions, Naukovyj visnyk Chernivets’kogo universytetu. Matematyka, 528 (2010), 104–110. (in Ukrainian)

13. O.V. Maslyuchenko, Decomposition of semi-continuous function as sum of two quasi-continuous functions and oscillation of almost continuous function, Mat. Stud., 35 (2011), 205–214. (in Ukrainian)

14. O.V. Maslyuchenko, D.P. Onypa, Limit oscillations of locally constant functions, Buk. Math. J., 1 (2013), №3–4, 97–99. (in Ukrainian)

15. O.V. Maslyuchenko, D.P. Onypa, Limit oscillations of continuous functions, Carp. math. publ., 7 (2015), № 2, 191–196. (in Ukrainian)

16. O.V. Maslyuchenko, D.P. Onypa, On cluster sets of continuous functions with values in locally arcwise connected spaces, Buk. Math. J., 3 (2015), №3–4, 127–132. (in Ukrainian)

17. O.V. Maslyuchenko, D.P. Onypa, Construction of functions with the given cluster sets, Colloquium Mathematicum (to appear) (http://arxiv.org/abs/1602.07118).

18. R. Engelking, General topology, PWN, Warsaw, 1977.

19. R. Arens, Extension of functions on fully normal spaces, Pacific Journal of Mathematics, 2 (1952), №1, 11-22.

20. K. Kuratowski. Topology. V.2, Moscow: Mir, 1969. (in Russian)

	The cluster sets of continuous functions (in Ukrainian)
Author	O. V. Maslyuchenko, D. P. Onypa denys.onypa@gmail.com, ovmasl@gmail.com Chernivtsi National University, Instytite of Mathematics, Academia Pomeraniensis in Slupsk
Abstract	Let $X$ be a metrizable space, $D$ be a boundary locally connected open subset of $X$ (i.e. for any $x\in \overline{D}\setminus D$ and for any neighborhood $U$ of $x$ there is a neighborhood $V$ of $x$ such that $V\subseteq U$ and $V\cap D$ is connected), $L$ be a closed subset of $\overline{D}\setminus D$ and $\Phi\colon L\multimap \overline{\mathbb R}$ be a multivalued mapping. We proved that there is a continuous function $f\colon D\to \mathbb R$ such that the cluster set $\overline{f}(x)=\Phi(x)$ for any $x\in L$ if and only if $\Phi$ is upper continuous compact-valued multifunction and $\Phi(x)$ is connected for any $x\in L$.
Keywords	cluster set; boundary locally connected set; usco mapping
DOI	doi:10.15330/ms.46.1.44-50
Reference	1. P. Painleve, Lecons sur la theorie analytique des equations differentielles professees a Stockholm, Paris, 1897. 2. P. Kostyrko, Some properties of oscillation, Math. Slovaca, 30 (1980), 157–162. 3. Z. Duszhynski, Z. Grande, S. Ponomarev, On the $\omega$-primitives, Math. Slovaka, 51 (2001), 469–476. 4. J. Ewert, S. Ponomarev, On the exsistance of $\omega$-primitives on arbitrary metric spaces, Math. Slovaca, 53 (2003), 51–57. 5. J. Ewert, S. Ponomarev, Oscillation and $\omega$-primiteves, Real Anal. Exchange, 26 (2001–2002), 687–702. 6. S. Kowalczyk, The $\omega$-problem, Diss. math. (Rozpr. mat.), 501 (2014), 1–55. 7. J. Borsik, On quasioscillation, Tatra Mountains Math. Publ., 2 (1993), 25–36. 8. O.V. Maslyuchenko, V.K. Maslyuchenko, The construction of a separately continuous function with given oscillation, Ukr. Math. J., 50 (1998), №7, 948–959. (in Ukrainian) 9. O.V. Maslyuchenko, The oscillation of quasi-continuous functions on pairwise attainable spaces, Houston Journal of Mathematics, 35 (2009), №1, 113–130. 10. O.V. Maslyuchenko, The construction of $\omega$-primitives: the oscillation of sum of functions, Mat. Visn. Nauk. Tov. Im. Shevchenka, 5 (2008), 151–163. (in Ukrainian) 11. O.V. Maslyuchenko The construction of $\omega$-primitives: strongly attainable spaces, Mat. Visn. Nauk. Tov. Im. Shevchenka, 6 (2009), 155–178. (in Ukrainian) 12. O.V. Maslyuchenko, Fluctuations of almost continuous functions, Naukovyj visnyk Chernivets’kogo universytetu. Matematyka, 528 (2010), 104–110. (in Ukrainian) 13. O.V. Maslyuchenko, Decomposition of semi-continuous function as sum of two quasi-continuous functions and oscillation of almost continuous function, Mat. Stud., 35 (2011), 205–214. (in Ukrainian) 14. O.V. Maslyuchenko, D.P. Onypa, Limit oscillations of locally constant functions, Buk. Math. J., 1 (2013), №3–4, 97–99. (in Ukrainian) 15. O.V. Maslyuchenko, D.P. Onypa, Limit oscillations of continuous functions, Carp. math. publ., 7 (2015), № 2, 191–196. (in Ukrainian) 16. O.V. Maslyuchenko, D.P. Onypa, On cluster sets of continuous functions with values in locally arcwise connected spaces, Buk. Math. J., 3 (2015), №3–4, 127–132. (in Ukrainian) 17. O.V. Maslyuchenko, D.P. Onypa, Construction of functions with the given cluster sets, Colloquium Mathematicum (to appear) (http://arxiv.org/abs/1602.07118). 18. R. Engelking, General topology, PWN, Warsaw, 1977. 19. R. Arens, Extension of functions on fully normal spaces, Pacific Journal of Mathematics, 2 (1952), №1, 11-22. 20. K. Kuratowski. Topology. V.2, Moscow: Mir, 1969. (in Russian)
Pages	44-50
Volume	46
Issue	1
Year	2016
Journal	Matematychni Studii
Full text of paper	pdf
Table of content of issue	html

The cluster sets of continuous functions (in Ukrainian)