Books like Problems and theorems in analysis by George Pólya

A comprehensive bridge from elementary calculus to advanced real and complex function theory, this textbook introduces the abstract reasoning characteristic of modern mathematical analysis.

Introduction to the Problems in Analysis outlines an elementary, one semester course which exposes students to both the process of rigor, and the rewards inherent in taking an axiomatic approach to the study of functions of a real variable. The aim of a course in real analysis should be to challenge and improve mathematical intuition rather than to verify it. The philosophy of this book is to focus attention on questions which give analysis its inherent fascination. Does the Cantor set contain any irrational numbers? Can the set of points where a function is discontinuous be arbitrary? Can the rational numbers be written as a countable intersection of open sets? Is an infinitely differentiable function necessarily the limit of its Taylor series? Giving these topics center stage, the motivation for a rigorous approach is justified by the fact that they are inaccessible without it.

A clear, sequential introduction to mathematical analysis covering continuity, differentiability and integration with numerous diagrams, margin notes, graded examples, and complete solutions for self-study or university courses.

A compilation of over one thousand problems from the Berkeley exams for mathematics students preparing for PhD programs, this book aids in problem-solving skills development across various mathematical areas, including real analysis, multivariable calculus, differential equations, metric spaces, complex analysis, algebra, and linear algebra.