Foundations of Real Analysis

Foundations of Real Analysis offers up a first course in real analysis aimed at advanced undergraduate students or new graduate students. The text covers the central topics of analysis, like continuity, differentiation, and integration, with a particular emphasis on set-theoretic and topological aspects of the real line, such as the Baire Category Theorem and the infinite-length Banach-Mazur games. It is richly illustrated and includes a wealth of interesting examples and counterexamples, such as Hilbert’s space-filling curves and Volterra’s non-integrable derivative. These mathematical spectacles aim to challenge the student’s preconceptions about the real line, while at the same time the main part of the text builds up a more well-founded intuition. Foundations of Real Analysis presents the core ideas of real analysis with intuition-driven arguments and visual appeal. The book connects analysis with other adjacent areas of mathematics, including important arguments and ideas from topology, measure theory, abstract algebra, descriptive set theory, and functional analysis. It aims to give the student a thorough and rigorous introduction to real analysis, leaning on the more intuitive and imaginative aspects of the subject, while also revealing some of the broader context of modern mathematics in which the subject is situated. This introductory course is designed not only for future analysts, but for anyone wanting to understand analysis and to sharpen their mathematical insight. The text is well suited to a two-semester university course, but can also be used for self-study by the curious reader.

• Introduces a clear and didactic understanding of essential concepts in real analysis, including compactness, differentiation, and integration
• Includes numerous illustrations, examples, and case studies, which provide clear explanations and additional context
• Aligns with commonly offered upper-level courses in real analysis and related mathematics programs
• Serves as a valuable resource for students and as a solid foundational material for early-stage researchers
• Offers online support, including additional homework resources, practice quizzes, and test banks