Real functions : current topics / Vasile Ene

This is a very concise book on real functions, specifically on those real functions which are relevant to a general theory of integration on the line. It is basically a reference book which could also be used by advanced graduate students as a text. The index consists of six pages, two columns each, of symbols with only an occasional word. It is the key to following the multitide of heavily symbolic proofs of theorems, which are frequently stated as inclusion relationships between classes of functions. The book is workable as a reference due to the fact that one can find the definitions of the symbols by using the index and then follow the proof. The last chapter contains many nicely done computer generated representations of useful examples of functions along with the proof that there are functions in certain classes which have the properties claimed of them. Chapter 1 consists of preliminary definitions and basic results of real analysis. They are usually not proved; the reader is given a reference to the proof in the literature. In Chapter 2 one finds the definitions of classes of functions (many of these are the creations of the author) and the inclusion relationships between them. In this chapter and those that follow, the proofs are usually provided. Chapter 3 deals with the results of Nina Bary on representations of continuous functions using compositions and sums. It contains her results along with some more recent ones. Chapter 4 deals with monotonicity theorems; that is, theorems giving sufficient conditions for a function to be monotone non-decreasing. These results are, naturally, useful in the construction of the primitives for a process of integration and for showing the uniqueness of the resulting integral. Chapter 5 presents a variety of integrals, including some classes of primitives defined by the author. It also presents and proves the equivalence of Perron, Denjoy and Henstock type integrals and presents more general classes of primitives. A substantial number of the generalizations are those of the author. The last chapter brings together examples of functions, frequently continuous, showing that certain inclusions among classes of functions are proper or that certain types of functions are possible. The visual representations of their graphs make use of a "box picture''; that is, the graph is constructed by replacing one set, consisting of rectangles and line segments, with the next. To construct the next set, each rectangle is replaced with a set resembling the contents of the previous rectangle. The intersection of the resulting sequence of sets is compact and thus is the graph of a continuous function. Again, many of these constructions are due to the author. (MathSciNet)