Foundations of Real Analysis

Foundations of Real Analysis: Expanding Horizons beyond the click 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. 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. 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 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 aims to give students 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.