@book {MATHEDUC.06518675, author = {Geveci, Tunc}, title = {Advanced calculus of a single variable.}, year = {2016}, isbn = {978-3-319-27806-3}, pages = {xii, 382~p.}, publisher = {Cham: Springer}, doi = {10.1007/978-3-319-27807-0}, abstract = {This volume is devoted to a thorough discussion of some basic concepts and theorems related to a beginning calculus course. The author is mainly interested to the development of the fundamental concepts of calculus, such as limits (with an emphasis on $\varepsilon$-$\delta$ definitions), continuity (including an appreciation of the difference between mere pointwise and uniform continuity), the derivative (with rigorous proofs of various versions of L'H\^opital's rule) and the Riemann integral (discussing improper integrals, comparison and Dirichlet tests). The content of the book under review is divided into the following six chapters: 1. Real numbers, sequences, and limits; 2. Limits and continuity of functions; 3. The derivative; 4. The Riemann integral; 5. Infinite series; 6. Sequences and series of functions. The first chapter begins with a quick review of the properties of the set of real numbers as an ordered field. The concept of the limit of a sequence and the relevant rules are discussed rigorously. The completeness of the field of real numbers is introduced as the existence of the limit of a Cauchy sequence. The least upper bound principle and the special nature of the convergence or divergence of a monotone sequence are also treated in Chapter 1. Even the notion of an infinite limit is also discussed here. In the next chapter, the author is interested in the development of the basic notions of continuity and limits of functions. The discussion is limited to functions defined on intervals. A central role in this development is played by the $\varepsilon$-$\delta$ definitions, which go back to the pioneering work of Cauchy. One of the highlights of the chapter is the intermediate value property, which has bearing on the definitions of basic inverse functions that figure prominently in beginning calculus. Chapter 3 is devoted to the notion of derivative and some of its main properties. Furthermore, convexity as a nice application of the mean value theorem is discussed in detail. Rigorous proofs of various versions of the L'H\^opital rule are also included in this chapter. Chapter 4 is concerned with the Riemann integral. Integrability criteria in terms of upper and lower sums and the oscillation of a function are discussed. A detailed discussion of improper integrals, including comparison and Dirichlet tests, is also included. Chapter 5 is a review of series of real numbers. A nice part in this chapter includes Cauchy-type criteria for the convergence of series. Chapter 6 discusses the convergence of sequences and series of real-valued functions on intervals. The distinction between mere pointwise and uniform convergence is emphasized with ample examples. The analyticity of functions defined via power series follows smoothly once the appropriate foundation involving uniform convergence is established. The chapter concludes with the definition of familiar special functions via power series. The presentation is thorough and clear with many comments on the historical context of the problems and concepts. Requiring only basic knowledge of elementary calculus, this book presents the necessary material for students and professionals in various mathematics-related fields, such as engineering, statistics, and computer science, to explore real analysis.}, reviewer = {Teodora-Liliana R\u{a}dulescu (Craiova)}, msc2010 = {I15xx}, identifier = {2016e.00802}, }