USCO and quasicontinuous mappings
In the preface of this book the following goal is set by the authors, who are noted experts on the topic: ``The purpose of our monograph is: \begin{itemize} \item[--] to introduce to nonspecialists the classes of usco and quasicontinuous mappings, \item[--] to present the most fundamental properties of these mappings, \item[--] to highlight their utility.'' \end{itemize} A first question arises: what is the main idea which led the authors to select, among various generalizations of continuity, in particular quasicontinuity and upper semicontinuity, and to discuss both together in the same book? Although these two notions are closely related via the continuity of single-valued maps, they are also, at first glance, rather distinct generalizations: separately quasicontinuous functions on product spaces are jointly quasicontinuous (and thus behave even better than continuous functions) whereas separately upper semicontinuous functions or multifunctions may even fail to be measurable (and thus behave worse); the sum of two (real) upper semicontinuous functions is usc (behaves like continuous functions) whereas the sum of two quasicontinuous functions may fail to be quasicontinuous (in fact each real cliquish function is the sum of two quasicontinuous functions). There are numerous answers to this first question, but each such answer assumes the existence of some common connecting leitmotif, forcing the subordinated structure of the book (the graph of content dependence should minimize the number of its leaf vertices). In the case of this book (the first monograph dedicated to this subject in the literature) the common idea connecting quasicontinuity and usco-ness are common applications, including such interesting areas as semitopological or paratopological groups, Glicksberg's theorem and others. The deepest interrelations between both generalizations of continuity are contained in profound interactions between usco multifunctions and their quasicontinuous selections (Theorem~1.3.10 on p.~23 due to J.~M. Borwein and W.~B. Moors and further developed by M.~Matejdes; see also [\textit{J.~Cao} and \textit{W.~B. Moors}, Real Anal. Exch. 31, No.~1, 63--71 (2006; Zbl 1097.54023)]). Quasicontinuity also plays an instrumental role in producing a densely defined selection with a natural continuity property relative to the domain [\textit{J.~R. Giles} and \textit{M.~O. Bartlett}, Set-Valued Anal. 1, No.~4, 365--378 (1993; Zbl 0805.54018)] and for the existence of usco multiselections for densely defined lower demicontinuous multifunctions [\textit{W.~B. Moors} and \textit{S.~Somasundaram}, Bull. Aust. Math. Soc. 65, No.~2, 307--313 (2002; Zbl 1028.54021)], which is mentioned on p.~92. From this perspective quasicontinuity and upper semicontinuity must be treated somewhat unsymmetrically: in fact, in this book quasicontinuous maps are single-valued (although the possibility of multivalued versions is mentioned), while upper semicontinuous maps are mainly compact multi-valued. Moreover, only circa 25\% of the text is devoted to quasicontinuity (Chapter~2 and some results scattered in other chapters). The book is structured -- in the reviewer's opinion -- rather from the point of view of applications of quasicontinuity in investigations of usco maps, than applications of upper semicontinuity in quasicontinuity. This, of course, is not any disadvantage, rather, it is even a valuable feature of the book. A second question is: what properties should be recognized as ``most fundamental'' to extract the leitmotif and ensure the unity of the text, and what must be intentionally omitted, in order to achieve coherence of ideas? From some point of view, the fact that each cliquish function may be represented as a sum of two quasicontinuous functions may appear as more fundamental than facts concerning fragmentable and Stegall spaces (p.~73). But, working in spaces without any algebraic structure, the sum of functions is unspeakable of and its omission is desirable to protect the text from growing exuberantly. Generally, the balance between the desire to cover all essential information and to stick to a uniform course is successful: this book (just like any other!) cannot be treated as a complete source of information concerning quasicontinuity, but this is rather its advantage, not its weakness (``It is in self-limitation that the master first reveals himself'' as Goethe said). On the other hand, it contains applications of quasicontinuity that may be new and unknown even to specialists working in this area, who are, however, not so familiar with multifunctions and topological groups. Chosen in that way the content is organized in four chapters: 1.~Usco mappings, 2.~Quasicontinuity, 3.~Applications of usco mappings, 4.~Topological properties of the space of usco mappings. Chapter 1 contains definitions, constructions and examples of usco multifunctions between topological spaces, the Kakutani-Glicksberg-Ky Fan theorem for cusco multifunctions (in the framework of Hausdorff locally convex linear topological spaces; without touching other generalizations), minimax theorems, characterizations and existence of minimal usco multifunctions, selected selection and multiselection theorems: Michael's selection theorem, the Jayne-Rogers selection theorem (and Baire-1 selection theorems in general), Fort's theorem on the fatness of the set of continuity points of compact-valued lower or upper semicontinuous multifunction defined on a Baire space, characterization of uscos into metric spaces and 21 exercises. Explanations on upper and Borel class of usco multifunctions (e.g., must an usco map belong to the lower Borel~1 class?) are, unfortunately, omitted in this book. Chapter 2 is devoted to the origin of the quasicontinuity notion, equivalent definitions in terms of semi-open sets, quasicontinuity of functions of two variables with separately quasicontinuous sections (Martin's generalization of Kempisty's Theorem, 2.2.4 on p.~61), and a detailed characterization of continuity points of quasicontinuous functions with values in metric spaces. Minimal usco maps appear in this chapter in the context of Stegall spaces. Constructions of non-Borel quasicontinuous maps and 9~exercises are provided. In the second half of this chapter facts concerning limits of quasicontinuous functions and the notion of equiquasicontinuity, interesting applications to semitopological and paratopological groups including Bouziad's theorem and Glicksberg's theorem, Ascoli-type theorems, and problems concerning the metrizability of the uniform space of quasicontinuous functions with values in metric space are presented. Even modern applications of quasicontinuity to compact Hausdorff admissible right topological (= CHART) groups (important in the theory of dynamical systems) are outlined. The role of Chapter~3 is to demonstrate the utility of the usco class in topology, approximation theory, differentiability of convex functions, variational calculus and differentiability of Lipschitz functions. Topological applications concern extensions of usco multifunctions and continuous functions and $K$-countably determined spaces. The next section shows how minimal usco multifunctions occur in the study of nearest and farthest points in (mainly strictly convex) normed spaces; basic information on Chebyshev sets are also included here. A complete and innovative (from the perspective of multifunctions) characterisation of Asplund and Radon-Nikodym spaces is given. The subsequent 26~pages are intended for the analysis of differentiability of convex functions (but not multifunctions). The notion of subdifferential appears, and its nature as a weak$^*$-cusco multifunction is exploited in Proposition~3.3.12; more general convex multifunctions and differential inclusions are rather beyond the scope of this book. Weak Asplund spaces and Gâteaux and Fréchet differentiablity of convex functions are sufficiently presented. Then variational theorems (including the important Stegall theorem on p.~185) and James's weak compactness theorem are proved and differentiability of Lipschitz (single-valued) functions is largely discussed, including Clarke's generalized subgradients (Theorem~3.6.8 on p.~200), D-representability and integrability. The chapter ends with 21~exercises. Quasicontinuity plays no role in that chapter. The subject of the final Chapter~4 is the topology of the space of usco and cusco multifunctions. Minimal usco and cusco multifunctions (and how they are related to quasicontinuous selections), densely continuous forms and their connections with minimal usco multifunctions, minimal cusco multiselections, quasicontinuity of upper envelopes of minimal cuscos, characterization of maps being simultaneously minimal usco and minimal cusco in terms of quasicontinuity, topologies on spaces of minimal uscos and minimal cuscos, Ascoli type theorems for related spaces, metrisability problems, countability and completeness-like properties of cuscos among many other interesting facts are described. This chapter contains 18~exercises. The Bibliography includes 314 publications, however the inclusion of a few more relevant books (e.g., [\textit{J.-P.~Aubin} and \textit{H.~Frankowska}, Set-valued analysis. Boston etc.: Birkhäuser (1990; Zbl 0713.49021)], [\textit{D.~Repovš} and \textit{P.~V. Semenov}, Continuous selections of multivalued mappings. Dordrecht: Kluwer (1998; Zbl 0915.54001)], [\textit{R.~S. Burachik} and \textit{A.~N. Iusem}, Set-valued mappings and enlargement of monotone operators. New York, NY: Springer (2008; Zbl 1154.49001)]) might have been helpful to the reader. This monograph is apparently the first in the mathematical literature explaining the relationship between upper semicontinuity of multifunctions and quasicontinuity of their selections. The scope, although nicely reduced and specialized, does not exclude using this advanced textbook as a basis for a monographic lecture on usco multifunctions and a source of condensed information (for readers having a background covering topology, functional analysis, measure theory, descriptive set theory), but it will also be useful for working mathematicians who want to understand the interplay between upper semicontinuity and quasicontinuity as two different, and not obviously interrelated, generalizations of continuity. The appearance of this well-constructed monograph should be welcomed with appreciation.
differentiability of Lipschitz functions
scientific article; zbMATH DE number 7438191