Mathematical Analysis Volume I

TL;DR

This textbook provides a comprehensive introduction to single-variable mathematical analysis, covering fundamental concepts, theorems, and classical examples, suitable for a two-semester course.

Contribution

It offers a detailed, structured presentation of core analysis topics with classical proofs and examples, serving as an educational resource.

Findings

Includes proofs of key theorems like intermediate value and Weierstrass approximation.

Features classical examples such as irrationality of e and non-analytic functions.

Covers advanced topics like power series and uniform convergence.

Abstract

This is the first volume of a textbook for a two-semester course in mathematical analysis. This first volume is about analysis of functions of a single variable. The topics covered include completeness axiom, Archimedean property, sequentially compact subsets of $\mathbb{R}$, limits of functions, continuous functions, intermediate value theorem, extreme value theorem, differentiation, mean value theorem, l'Hopital's rule, Riemann integrals, improper integrals, elementary transcendental functions, sequences and series of numbers, infinite products, sequences and series of functions, uniform convergence, power series, Taylor series and Taylor polynomials. At the end of the book, we include some classical examples such as the irrationality of the number $e$, the existence of a non-analytic infinitely differentiable function, the existence of a nowhere differentiable continuous function. The book is concluded with the proof of the Weierstrass approximation theorem.