Classical Analysis of Real-Valued Functions

Divided into two self-contained parts, this textbook is an introduction to modern real analysis. More than 350 exercises and 100 examples are integrated into the text to help clarify the theoretical considerations and the practical applications to differential geometry, Fourier series, differential equations, and other subjects. The first section of Classical Analysis of Real-Valued Functions covers the theorems of existence of supremum and infimum of bounded sets on the real line and the Lagrange formula for differentiable functions. Applications of these results are crucial for classical mathematical analysis, andmany are threaded through the text. In the second part of the book, the implicit function theorem plays a central role, while the Gauss-Ostrogradskii formula, surface integration, Heine-Borel lemma, the Ascoli-Arzelà theorem, and the one-dimensional indefinite Lebesgue integral are also covered. This book is intended for students in the first and second years of classical universities majoring in pure and applied mathematics, but students of engineering disciplines will also gain important and helpful insights. It is appropriate for courses in mathematical analysis, functional analysis, real analysis, and calculus and can be used for self-study as well.…