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This volume presents an expanded and refined exploration of the mathematical analysis of univariate real functions, developed from a series of eleven postgraduate lectures on Real Analysis first delivered in September 2016. Building upon the foundations of a previous textbook, this extended edition deepens the discussion of core analytical concepts and introduces several new sections-marked with an asterisk (_)-that address advanced topics not covered in the original lectures. Structured into eleven comprehensive chapters and an appendix, the book offers a rigorous yet accessible path through the essential tools, theorems, and techniques of real analysis. It is designed for graduate students, instructors, and researchers seeking a solid and modern treatment of the subject. Serving both as a standalone reference and a prelude to the forthcoming Mathematical Analysis I , which will further expand the theory with six new chapters and three appendices, this work continues the author's pursuit of clarity, depth, and mathematical precision.