A Course in Analysis - Volume I: Introductory Calculus, Analysis of Functions of One Real Variable

Volume 1 covers the content of two typical modules in undergraduate course for mathematics: Part 1: Introductory calculus Part 2: Analysis of functions of one variableThese two parts are divided into 32 chapters. Part 1 begins with an overview of a set of real numbers: rational and irrational. This is accompanied by a discussion of arithmetic rules, inequalities, absolute values. Doing this, the authors, in a clever way, introduce elements of a set theory using Cantor's approach. This is followed by the definition of a function and discussion of operations on functions. From this they move to differential and integral calculus and their applications.Part 2 contains rigorous proofs of theorems from Part 1. In particular, they introduce the order structure on a real line using axioms of an ordered field. It is worth mentioning that Part 2 contains elements of topology obviously on real line and a good insight on convex functions. The convex functions relate Analysis to linear spaces equipped with norm. Finally, to close Part 2 the authors supply a number of Appendices, among them Appendices on logic, set theory and Peano axioms.Remarkably, Volume 1 contains 360 problems with complete solutions.