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Modern Real Analysis - ebook/pdf
Modern Real Analysis - ebook/pdf
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Wydawca: Wydawnictwo Uniwersytetu Łódzkiego Język publikacji: Angielski
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Kategoria: ebooki >> naukowe i akademickie >> matematyka
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W monografii zaprezentowano rezultaty badań związanych ze współczesnymi trendami badawczymi w teorii funkcji rzeczywistych. Takiej tematyce są poświęcone międzynarodowe konferencje z teorii funkcji rzeczywistych organizowane od wielu lat przez polsko-słowackie środowisko matematyczne. W publikacji znajduje się krótki rys historyczny tych konferencji, a także biografie naukowe Profesorów Jubilatów – Romana Gera, Jacka Jędrzejewskiego i Zygfryda Kominka, którym dedykowana jest książka.

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Jacek Hejduk – University of Łódź, Faculty of Mathematics and Computer Science Department of Real Functions, Banacha 22, PL-90-238 Łódź Ryszard J. Pawlak – University of Łódź, Faculty of Mathematics and Computer Science Department of Methodology Teaching Mathematics, Banacha 22, PL-90-238 Łódź Stanisław Kowalczyk, Małgorzata Turowska – Pomeranian University in Słupsk Institute of Mathematics, Arciszewskiego 22b, PL-76-200 Słupsk Marek Balcerzak, Artur Bartoszewicz, Szymon Głąb, Jacek Hejduk, Grażyna Horbaczewska, Jacek Jędrzejewski, Stanisław Kowalczyk, Tomasz Natkaniec, Jurij Povstenko, Franciszek Prus-Wiśniowski Marian Przemski, Elżbieta Wagner-Bojakowska, Wojciech Wojdowski REVIEWERS INITIATING EDITOR Damian Rusek TYPESETTING Stanisław Kowalczyk, Małgorzata Turowska PHOTO Małgorzata Terepeta COVER DESIGN Katarzyna Turkowska Cover image: © Depositphotos.com/Glass cubes Printed directly from camera-ready materials provided to the Łódź University Press This publication has been financed by the Faculty of Mathematics and Computer Science © Copyright by Authors, Łódź 2015 © Copyright for this edition by Uniwersytet Łódzki, Łódź 2015 Published by Łódź University Press First Edition W.07154.15.0.K Printing sheets 14,125 ISBN 978-83-7969-663-5 e-ISBN 978-83-7969-955-1 Łódź University Press 90-131 Łódź, 8 Lindleya St www.wydawnictwo.uni.lodz.pl e-mail: ksiegarnia@uni.lodz.pl tel. (42) 665 58 63 Dedicated to Professors Roman Ger, Jacek Jędrzejewski, Zygfryd Kominek Contents Chapter 1 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Preface Jacek Hejduk, Stanisław Kowalczyk, Ryszard J. Pawlak and Małgorzata Turowska 7 Chapter 2 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 11 Quasicontinuous functions with small set of discontinuity points Ján Borsík Chapter 3 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 27 Convolution operators on some spaces of functions and distributions in the theory of circuits Andrzej Borys, Andrzej Kami´nski and Sławomir Sorek Chapter 4 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 45 Continuity connected with ψ-density Małgorzata Filipczak and Małgorzata Terepeta Chapter 5 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 61 On equivalence of topological and restrictional continuity Katarzyna Flak and Jacek Hejduk Chapter 6 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 69 On fields inspired with the polar HSV − RGB theory of Colour Ján Haluška 6 Contents Chapter 7 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 89 Generalized (topological) metric space. From nowhere density to infinite games Ewa Korczak-Kubiak, Anna Loranty and Ryszard J. Pawlak Chapter 8 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 105 Path continuity connected with density and porosity Stanisław Kowalczyk and Małgorzata Turowska Chapter 9 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 127 Topological similarity of functions Ivan Kupka Chapter 10 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 147 On the Darboux property of derivative multifunction Gra˙zyna Kwieci´nska Chapter 11 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 161 On extensions of quasi-continuous functions Oleksandr V. Maslyuchenko and Vasyl V. Nesterenko Chapter 12 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 185 Weak convergence with respect to category Władysław Wilczy´nski Chapter 13 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 193 New properties of the families of convergent and divergent permutations - Part I Roman Wituła, Edyta Hetmaniok and Damian Słota Chapter 14 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 213 New properties of the families of convergent and divergent permutations - Part II Roman Wituła Chapter 1 Preface JACEK HEJDUK, STANISŁAW KOWALCZYK, RYSZARD J. PAWLAK, MAŁGORZATA TUROWSKA The first conference on Real Functions Theory had been organized by Mathe- matical Institute of the Slovak Academy of Sciences in Bratislava since 1971. Since 1972 conferences were organized by other institutions every second year and since 2005 a Polish group of mathematicians joint to organize con- ferences every second year. So since 2004 International Summer Conferences on Real Functions Theory held every year. The latter conference is XXIX, one in the series that consists of: (1) (2) (3) (4) (5) (6) (7) (8) (9) (10) (11) (12) (13) (14) 1971 — Modra - Harmónia 1972 — ˇCingov 1974 — L’ubochˇna 1976 — Modra - Harmónia 1978 — Modra - Harmónia 1980 — Trnava 1982 — Dolné Orešany 1984 — Kamenný Mlyn 1986 — Richˇnava 1988 — Dubník 1990 — Dubník 1992 — Dubník 1994 — Liptovský Ján 1996 — Liptovský Ján 8 Jacek Hejduk, Stanisław Kowalczyk, Ryszard J. Pawlak, Małgorzata Turowska (15) (16) (17) (18) 1998 — Liptovský Ján 2000 — Liptovský Ján 2002 — Stará Lesná 2004 — Stará Lesná First 18 conferences were organized by Slovak Academy of Sciences. Since 2005 in odd years the International Summer Conferences on Real Functions Theory are organized by Pomeranian Academy in Słupsk, University of Łód´z, Łód´z Technical University and University of Computer Sciences and Skills, and in even years the International Summer Conferences on Real Functions Theory are organized by Slovak Academy of Sciences. (19) (20) (21) (22) (23) (24) (25) (26) (27) (28) (29) 2005 — Rowy (Poland) 2006 — Liptovský Ján (Slovakia) 2007 — Niedzica (Poland) 2008 — Stará Lesná (Slovakia) 2009 — Niedzica (Poland) 2010 — Liptovský Ján (Slovakia) 2011 — Złoty Potok (Poland) 2012 — Stará Lesná (Slovakia) 2013 — Niedzica (Poland) 2014 — Stará Lesná (Slovakia) 2015 — Niedzica (Poland) Thus in 2015 we celebrate 10th anniversary since Pomeranian Academy in Słupsk, University of Łód´z and Łód´z University of Technology joined Slovak Academy of Science and started organizing the International Summer Confer- ences on Real Functions Theory every odd year. As far, six conferences orga- nized by Polish institutions were held, four in Niedzica in Pieniny Mountains, one in Rowy near Baltic Sea and one in Złoty Potok in Central Poland. International Summer Conferences on Real Functions Theory has a wide group of regular participants. Three of them, Professor Roman Ger, Professor Jacek J˛edrzejewski and Professor Zygfryd Kominek, celebrate this year seven- tieth birthday anniversary. Professor Roman Ger was born at July 30, 1945. He received his master’s degree in 1968 at a branch of Jagiellonian University in Katowice. In 1971 he received Ph. D. degree and habilitation in 1976, both from Silesian University in Katowice. He became a full professor in 1990. Professor Roman Ger started research under the supervising of Professor Marek Kuczma, the founder of Polish school of functional equations. From 1976 till now he works at Silesian University. From 2005 to 2008 he was Deputy Director and from 2008 to 2012 1. Preface 9 he was Director of the Mathematical Institute. Between 1990 and 1992 he was a Vice Chan- cellor of Silesian University. Simultaneously, he worked in Academy of Jan Długosz in Cz˛estochowa in years 1981-1987, 1991-2010 and in Silesian University of Technology in Gliwice in 1988-1991. He was a visiting pro- fessor at the University of Waterloo, Waterloo, Ontario, Kanada, 1979; University of Central Florida, Orlando, Floryda, USA, 1987-88; Karl-Franzens-Universität, Graz, Austria, 1994; Universitaät Bern, Berno, Szwajcaria, 1995. In years 1984-2000 he led, founded by Professor Marek Kuczma, the seminar on Functional Equations. In 1981 he founded and leads till now the seminar on Functional Equations and Inequalities. His scientific output contains more than 110 scientific papers, mainly de- voted to functional equations and inequalities. He gave more than 50 lec- tures among the other Universities in USA, Canada, Germany, Italy, Austria, Switzerland, Spain, Greece, Hungary, Czech Republic, Denmark, Israel, China Republic and Venezuela. He was a supervisor of 15 Ph.D. students and has served as a referee for 26 Ph.D. thesis, 8 habilitation dissertations and 18 ap- plications for professor nomination. Professor Jacek J˛edrzejewski was born at August 6, 1945. He received his master’s degree in 1969, Ph. D. degree in 1974 and habilitation degree in 1984, all of them from University of Łód´z. He worked at the University of Łód´z in years 1969-1984. He was a Senior Lecturer in Rivers State University of Science and Technology, Port Harcourt, Nigeria, 1984-1988. After his return to Poland he worked at Higher Pedagogical School in Bydgoszcz, Pomeranian Academy in Słupsk 1989-2005, University of Computer Sci- ences and Skills 1999-2011 and Academy of Jan Długosz in Cz˛estochowa 2005-2015. From 1988 to 1993 he was a Deputy Di- rector and from 1993 to 1995 he was the Director of the Mathematical Institute at Higher Pedagogical School in Bydgoszcz. From 1989 to 1990 he was a Vice Dean and from 1990 to 1993 he was a Vice Chancellor at Higher Pedagogical School in Bydgoszcz. He also was the Director of the Mathematical Institute at Pomeranian Academy in Słupsk 1999-2005. 10 Jacek Hejduk, Stanisław Kowalczyk, Ryszard J. Pawlak, Małgorzata Turowska His scientific output contains more than 40 papers, mainly devoted to func- tion theory and topology. He was the supervisor of 3 Ph.D. students and has served as a referee for 4 Ph.D. thesis. In 1997 Professor Jacek J˛edrzejewski organized the first International Con- ference on Real Functions Theory in Ustka. Next conferences took place in 2001 in Łeba and in 2003 in Rowy. In 2005 these conferences evolved into International Summer Conferences on Real Functions Theory. Professor J˛e- drzejewski was the president of all International Summer Conferences on Real Functions Theory organized by Polish Universities. Professor Zygfryd Kominek was born at November 8, 1945. He received his master’s degree in 1968 at a branch of Jagiellonian University in Katowice. In 1974 he received Ph. D. degree from Silesian University in Katowice and in 1991 he received habilitation degree from Warsaw University of Technology. He became a full professor in 2008. From 1976 till now he works at Silesian University. From 1992 to 1996 and from 2011 till now he was a Deputy Director and from 1996 to 2002 he was the Director of Mathe- matical Institute. Simultaneously, he worked in Academy of Jan Długosz in Cz˛estochowa 1992-1998, 2004-2006, in Łód´z University of Technology branch in Bielsko-Biała 1991-2001, in The University of Bielsko-Biała 2001-2003 and in the Katowice Institute of Information Technologies 2006-2011. His scientific output contains more than 70 papers, mainly devoted to condi- tional functional equations, convex functions and systems of functional equa- tions. He was the supervisor of 5 Ph.D. students and has served as a referee for 16 Ph.D. thesis. The presented monograph Modern Real Analysis is dedicated to the men- tioned anniversaries. It contains several chapters, written by frequent partici- pants of International Summer Conferences on Real Functions Theory, where their actual research topics are presented. Chapter 2 Quasicontinuous functions with small set of discontinuity points JÁN BORSÍK 2010 Mathematics Subject Classification: 54C30, 54C08. Key words and phrases: quasicontinuous functions, cliquish functions, points of continuity, set of first category, set of measure zero. 2.1 Introduction The definition of quasicontinuity for real functions of real variable was given in [34] by S. Kempisty. Nevertheless, R. Baire in his work [1] has shown that a function of two variables continuous at each variable is quasicontinuous. An independent definition was given by W. W. Bledsoe [2] in 1952 under the name neighborly function. S. Marcus in [49] proved that the notions of neighborly and quasicontinuous functions are equivalent and he developed further proper- ties of quasicontinuous functions. He showed that quasicontinuous functions need not be (Lebesgue) measurable and for each countable ordinal α there is a quasicontinuous function in the Baire class α + 1 which does not belong to Baire class α. N. Levine in [44] introduced the notion of semi-continuous function as a function for which the inverse image of every open set is a semi-open set (a set A is semi-open if A is a subset of the closure of the interior of A). A. Neubrun- nová in her paper [53] has shown that the notions of quasicontinuity and semi- continuity in the sense of Levine are equivalent. Z. Grande in [33] has shown 12 Ján Borsík that a function f is quasicontinuous if and only if the graph of the function f restricted to the set of all continuity points of f is dense in the graph of f . A fundamental result concerning continuity points is due to N. Levine [44] for functions with values in a second countable space (and for functions with values in a metric space [53]) is that the set of discontinuity points of a quasi- continuous function is small. Theorem 2.1. Let X be a topologocal space and let Y be a second countable space ([44]) or let Y be a metric space ([53]). If f : X →Y is a quasicontinuous function then the set of discontinuity points is of first category. So, quasicontinuous functions have the Baire property. On the other hand, if X = R2 [19] or if X is a Baire pseudometrizable space space without isolated points (or X is a Baire resolvable perfectly normal locally connected space) [5] or X is a hereditarily separable perfectly normal Fréchet-Urysohn space [50], then for each Fσ -set A of first category there is a quasicontinuous func- tion f : X → R such that A is the set of all discontinuity points of this function. Points of quasicontinuity were characterized in [45]. Quasicontinuous func- tions were investigated very intensively. We recommend a survey [52] pub- lished in 1988 with more than 120 references. 2.2 Basic definitions Let R, Q and N be the set of all real, rational and positive integer numbers, respectively. For a set A ⊂ R denote by IntA and ClA the interior and the closure of A, respectively. Recall that a function f : X → Y (X and Y are topological spaces) is said to be quasicontinuous at a point x if for each neighbourhood U of x and each neighbourhood V of f (x) there is an open nonempty set G ⊂ U such that f (G) ⊂ V [34]. A function f : X → Y (X is a topological space and (Y,d) is a metric space) is said to be cliquish at a point x ∈ X if for each neighbourhood U of x and each ε 0 there is an open nonempty set G ⊂ U such that d( f (y), f (z)) ε for each y,z ∈ G [63]. H. P. Thielman introduced cliquish functions: Denote by C( f ), D( f ), Q( f ) and K( f ) the set of all continuity, disconti- nuity, quasicontinuity and cliquishness points of f , respectively. A function f is quasicontinuous (cliquish) if Q( f ) = X (K( f ) = X). Further, denote by C (X,Y ), Q(X,Y ) and K (X,Y ) (or briefly C , Q and K ) the family of all
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