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While many real analysis texts prioritize brevity over clarity, this volume focuses on the student’s conceptual journey.
The text begins with a rigorous construction of the Real Number System from the Field and Order Axioms, moving through the topology of R, sequences, and series. It provides a thorough treatment of continuity and differentiation, including the topological characterization of continuity. A key feature is the rigorous development of the Riemann Integral via Darboux sums, leading to the Lebesgue Criterion. The final chapters transition into Pointwise vs. Uniform Convergence and an introduction to Metric Spaces. The book uses a modern layout with color-coded theorem environments and visual proofs to aid learning.
It is designed as a core text for a one-semester Real Analysis course, specifically for students who find standard texts too opaque.
Paul L. Dayao is a faculty member at the Ateneo de Manila University with over a decade of teaching experience to his writing. His research interests lie in Real Analysis and Functional Analysis. He has recently published work on integration theory in the Missouri Journal of Mathematical Sciences (MJMS) and currently has seven other papers under review across various journals.
In addition to his research, he is an active textbook author. He has a college calculus textbook forthcoming with C&E Publishing, and two other manuscripts currently under consideration at major publishers: Fundamentals of Classical Banach Space Theory (under review at Springer) and Introductory Course in Functional Analysis (currently being considered by an editor at CRC Press/Taylor & Francis). He is also finalizing his PhD application in Measure Theory at the University of St Andrews. As a dedicated educator, he specializes in creating “teaching-led” mathematical materials that bridge the gap between intuition and rigor.