Limits of sequences of Darboux-like mappings

O. Karlova

We call a mapping

$f\colon X\to Y$ an

$l$ -Darboux mapping if the image of any arcwise connected subset of

$X$ is connected. We prove that the class of

$l$ -Darboux

$F_\sigma$ -measurable mappings of a topological space to a metric space is closed with respect to uniform limits.

1. A. Bruckner, Differentiation of real functions, 2nd ed., Providence, RI: American Mathematical Society, 1994, 195 p.

2. R. Gibson, T. Natkaniec, Darboux-like functions, Real Anal. Exchange, 22 (1996-97), №2, 492–533.

3. O. Karlova, V. Mykhaylyuk, On Gibson functions with connected graphs, Math. Slovaca, 63 (2013), №3.

4. K. Kellum, Functions that separate

$X\times\mathbb R$ , Houston J. Math., 36 (2010), 1221–1226.

5. K. Kuratowski, Topology, V.1, M.: Mir, 1966. (in Russian)

6. A. Lindenbaum, Sur quelques proprieties des fonctions de variable reelle, Ann. Soc. Math. Polon., 6 (1927), 129–130.

7. H. Pawlak, R.J. Pawlak, On weakly Darboux functions and some problem connected with the Morrey monotonicity, Journal of Applied Analysis, 1 (1995), №2, 135–144.

	Limits of sequences of Darboux-like mappings
Author	O. Karlova Maslenizza.ua@gmail.com Yuriy Fedkovych Chernivtsi National University
Abstract	We call a mapping $f\colon X\to Y$ an $l$ -Darboux mapping if the image of any arcwise connected subset of $X$ is connected. We prove that the class of $l$ -Darboux $F_\sigma$ -measurable mappings of a topological space to a metric space is closed with respect to uniform limits.
Keywords	uniform limit; Darboux function; $F_\sigma$ -measurable function; weakly Gibson function
Reference	1. A. Bruckner, Differentiation of real functions, 2nd ed., Providence, RI: American Mathematical Society, 1994, 195 p. 2. R. Gibson, T. Natkaniec, Darboux-like functions, Real Anal. Exchange, 22 (1996-97), №2, 492–533. 3. O. Karlova, V. Mykhaylyuk, On Gibson functions with connected graphs, Math. Slovaca, 63 (2013), №3. 4. K. Kellum, Functions that separate $X\times\mathbb R$ , Houston J. Math., 36 (2010), 1221–1226. 5. K. Kuratowski, Topology, V.1, M.: Mir, 1966. (in Russian) 6. A. Lindenbaum, Sur quelques proprieties des fonctions de variable reelle, Ann. Soc. Math. Polon., 6 (1927), 129–130. 7. H. Pawlak, R.J. Pawlak, On weakly Darboux functions and some problem connected with the Morrey monotonicity, Journal of Applied Analysis, 1 (1995), №2, 135–144.
Pages	132-136
Volume	40
Issue	2
Year	2013
Journal	Matematychni Studii
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