@book {MATHEDUC.06268767,
author = {Bashirov, Agamirza E.},
title = {Mathematical analysis fundamentals.},
year = {2014},
isbn = {978-0-12-801001-3},
pages = {xiii, 348~p.},
publisher = {Amsterdam: Elsevier},
abstract = {This book, based on a combination of the author's lecture notes from different courses on analysis from graduate and undergraduate levels, is intended to the readers who are familiar with differential and integral calculus and would like to receive a stronger mathematical background and rigorous fundamentals in analysis. Therefore, it is suggested as a real analysis textbook for second- or third-year students who have studied differential and integral calculus their first year. It may also be served as a real analysis textbook for first-year students of mathematical departments. An important motivation for the author to write such a book was his effort to offer a proper transition rate from elementary calculus to rigorous analysis. The book invites the reader to think abstractly. Each chapter has a list of exercises, which are mainly designed to clarify the meanings of the definitions and theorems, to force understanding the proofs, and to call attention to points in the proofs that might be overlooked. In Chapter~1 the author discusses basic elements of the set theory including an axiomatic approach. Another feature of this chapter is an emphasis of techniques of proofs. Chapter~2 introduces the numbers from set-theoretic concepts and extends the subject to cardinality. The relationship between two fields of numbers, rational and real, is emphasized. Convergence is discussed in the following two chapters. Chapter~3 deals with convergence in the system of real numbers with the attention to the fact that numerical series are just another, but often more convenient, form of numerical sequences. A deeper understanding of the concept of convergence comes with metric spaces, which may or may not have the features of the system of real numbers. These topics are discussed in Chapter~4. The following two chapters are devoted to continuity. Chapter~5 covers the definition and relationship of continuity and different concepts such as limit, compactness, and connectedness. Chapter~6 gives a general look at continuity and deals with the space of continuous functions and its different features as a metric space. Chapter~7 contains a standard presentation of differentiation with an emphasis on its relationship to continuity and mean-value theorems. It also contains two sections for further reading that deal with differential equations and the space of differentiable functions. In Chapter~8, the functions of bounded variation are studied. While many textbooks cover this subject by considering only monotone functions, this book goes beyond, discussing discrete and continuous functions of bounded variation and a space of these functions. Integration is presented in the next two chapters. Chapter~9 contains standard explanatory material (Riemann integration), while Chapter~10 gives an overall view to the integration of functions of a single variable. The reader can find here other definite integrals like Riemann-Stieltjes, Kurzweil-Henstock and Lebesgue integral. The creation of functions and transcending the capacity of rational functions are presented in Chapter~11. An interesting feature of this chapter is an introduction to multiplicative calculus that breaks down the thought about absoluteness of familiar ordinary calculus. Finally, in Chapter~12 trigonometric series and integrals are treated.},
reviewer = {Petr Gurka (Praha)},
msc2010 = {I15xx},
identifier = {2015a.00702},
}