About This Course

Calculus tells you that a continuous function on a closed interval has a maximum, that a function with zero derivative is constant, and that integrating a derivative recovers the function. It rarely tells you why. This book supplies the reasons. It rebuilds the calculus of one real variable from a single property of the real numbers, proves each of its main theorems with stated hypotheses, and shows by counterexample what happens when a hypothesis is dropped.

The book is written for a reader working alone who has finished a first calculus course such as Calculus and has met proofs before: direct proof, contradiction, contraposition, and induction, at the level of Discrete Mathematics. You should be comfortable with sets, functions, and the quantifiers “for every” and “there exists”. No earlier exposure to analysis is assumed. What is assumed is patience with statements in which the order of the quantifiers matters, because nearly every idea in the subject is a statement of that kind.

By the end you should be able to do four things without notes. State the definitions of supremum, limit, Cauchy sequence, open and closed set, compact set, continuity, uniform continuity, derivative, Riemann integral, and uniform convergence, with every quantifier in the right place. Prove the theorems that connect them, among them the Bolzano–Weierstrass theorem, the Heine–Borel theorem, the Extreme and Intermediate Value Theorems, the Mean Value Theorem, the Fundamental Theorem of Calculus, and the theorems that allow a limit to pass through an integral or a derivative. Produce a counterexample when a hypothesis is removed. And, facing a new claim, decide which tool applies: an order argument with suprema, a sequential argument, a compactness argument, or a uniform estimate.

A word about the elementary functions. Polynomials, rational functions, and roots are constructed inside the book. The functions \(\sin\) , \(\cos\) , \(\exp\) , and \(\log\) appear in examples and exercises before they are constructed, with the properties you know from calculus: their derivatives, the bounds \(\lvert\sin x\rvert\le 1\) and \(\lvert\cos x\rvert\le 1\) , and the functional equations of the exponential and the logarithm. No theorem of the book depends on them. Chapter 7 constructs the logarithm as an integral and the exponential as its inverse, and Chapter 8 recovers the exponential as a power series and indicates how the sine and cosine are built the same way, which closes the gap.

Notation is standard. \(\mathbb N=\{1,2,3,\dots\}\) , \(\mathbb Z\) , \(\mathbb Q\) , and \(\mathbb R\) are the natural numbers, integers, rationals, and reals. A sequence is written \((x_n)\) , and \((x_{n_k})\) is a subsequence. Intervals are \((a,b)\) , \([a,b]\) , \([a,b)\) , and so on. The letter \(\varepsilon\) always stands for a positive real number that someone else chooses and you must cope with.

Each chapter ends with exercises, most of which ask for a proof. Write your proofs out in full sentences; an argument you can only gesture at is not yet a proof. The companion solutions give complete proofs for some exercises, graduated hints for others, and short answers for a few. Several exercises present a plausible argument that is wrong and ask you to find the flaw. Take those seriously, because the wrong arguments of analysis are the ones that calculus intuition suggests. The cumulative review at the end does not say which chapter each problem belongs to; choosing the tool is part of the problem.