Dover Books on Mathematics: Introduction to Analysis

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This well-written text offers excellent instruction in basic real analysis, providing a solid foundation for advanced studies in complex analysis, differential equations, integration theory, and general topology. While a year of calculus is the nominal prerequisite, the material assumes only the axioms of the real number system. Its clarity and simplicity make it accessible to a broad range of students from various fields. Chapters cover set theory, the real number system, metric spaces, continuous functions, differentiation, Riemann integration, limit operations, successive approximations, partial differentiation, and multiple integrals. After introductory material on basic set theory and key properties of the real number system, the author presents a rigorous discussion of metric spaces and continuous functions, addressing open and closed sets, limits, continuity, and convergent sequences of points and functions. Subsequent chapters smoothly integrate elementary calculus with more advanced topics, such as multivariable calculus and existence theorems. Exercises range from easy problems to challenging ones, featuring interesting examples, counterexamples, and advanced results. This text is suitable for a one- or two-quarter or one-semester undergraduate course and has been refined through extensive classroom use, making it an unusually accessible introductory resource.