FROM CALCULUS TO ANALYSIS: A DEEP DIVE: An Illustrated Workbook in Proof, Pathology, Measure, Function Spaces, and Approximation

A Rigorous Problem-Solving Bridge from Calculus to Real Analysis

From Calculus to Analysis: A Deep Dive is a 300-problem workbook that guides students across the critical transition from computational calculus to formal, proof-based real analysis. Rather than lecturing at you, this book puts you to work: each problem pairs a fill-in-the-steps exercise with a fully worked solution key, building proof-writing skill through repetition, structure, and visual intuition.

Across four parts and twenty chapters, you will move from logic and set theory through the construction of the real numbers, then explore the classic pathological functions that reshaped analysis, advanced integration theory, and the function-space theorems that define modern mathematical analysis. Every chapter follows a consistent format: concept focus, a diagram description, a guided exercise with blanks to complete, and a complete blockquoted solution key.

What’s Inside

• Part I — The Transition to Proof: logic, quantifiers, and proof techniques; set theory and functions; induction and the well-ordering principle; construction of the reals via Dedekind cuts; supremum and infimum puzzles.
• Part II — Pathological Functions and Counterexamples: Thomae’s popcorn function; Dirichlet’s nowhere-continuous function; the Weierstrass continuous nowhere-differentiable function; the Cantor set and the devil’s staircase; Volterra’s function.
• Part III — Advanced Integration Theory: Riemann-Stieltjes integrals; functions of bounded variation; sets of measure zero; Lebesgue’s criterion for Riemann integrability; an introduction to Lebesgue measure.
• Part IV — Function Spaces and Approximations: the space of continuous functions C[a,b]; the uniform metric and completeness; the Arzelà-Ascoli theorem; the Stone-Weierstrass theorem; polynomial approximation via the Weierstrass Approximation Theorem.

How This Workbook Helps You Learn

Each of the 300 problems follows a fixed structure: a one-sentence concept focus, a visual description of the scenario, a step-by-step exercise with key steps left blank for you to complete, and a fully solved answer set apart in a clearly marked solution key. This fill-in-the-steps method forces active engagement with each proof rather than passive reading, while the visual descriptions build geometric intuition around otherwise abstract counterexamples and theorems.

Who This Book Is For

• Students taking their first course in real analysis after completing calculus
• Self-learners who want a structured, problem-driven path through classical counterexamples and advanced analysis
• Mathematics and physics majors preparing for graduate-level analysis, measure theory, or functional analysis
• Instructors seeking a ready-made bank of fill-in-the-steps proof problems with complete solutions

This is a workbook built for active study — 300 problems meant to be worked through step by step, chapter by chapter, from your first quantifier to the Stone-Weierstrass theorem.

Scroll up and get your copy to master the deep dive from calculus into rigorous real analysis.