Univariate Real Analysis

Preliminary version of a book on univariate real analysis, with 14 chapters and 2 appendices. 1. Real numbers; 2. Limits of real sequences; 3. Series; 4. Limits of real functions. 5. Elementary functions; 6. Continuous functions; 7. Derivatives; 8. Mean value theorems; 9. Taylor polynomials ; 10. Real analytic functions; 11. Newton integral; 12. Riemann integral; 13. Henstock-Kurzweil integral; 14. Applications of integrals; A. Auxiliary notions and notation; B. Solutions to exercises

💡 Research Summary

Martin Klazar’s “Univariate Real Analysis” is a comprehensive, preliminary textbook that expands a traditional first‑year analysis course into a full‑scale monograph consisting of fourteen chapters and two appendices. Written between 2024 and 2026, the work is intended for computer‑science students at the Faculty of Mathematics and Physics, Charles University, but its depth and breadth make it suitable for any undergraduate or early‑graduate audience interested in a rigorous treatment of one‑variable real analysis.

The book opens with a meticulous construction of the real numbers. Chapter 1 presents three classical constructions—Cantor–Merey, Dedekind cuts, and decimal expansions—side by side, proving their equivalence (Theorem 1.6.19) and the uncountability of ℝ via Cantor’s diagonal argument. Early on, functions and their congruence are defined (Definitions 1.1.2–1.1.3), establishing a uniform language for later sections.

Chapter 2 develops the theory of sequences and limits. Beyond the usual ε‑δ approach, Klazar introduces an “infinite element” R* together with a shift axiom, allowing a systematic algebra of limits that includes infinite values (Theorem 2.1.6). A striking result (Theorem 2.2.16) gives a precise duality: a sequence fails to converge iff it possesses two subsequences with distinct limits. The chapter also contains a modern treatment of Fekete’s lemma (additive and multiplicative versions) and shows how these lemmas feed into combinatorial applications later in the book.

Chapter 3 is perhaps the most original part of the manuscript. The author defines AK (absolutely convergent) series as maps r : X → ℝ on at most countable index sets X, with a uniform bound on finite partial sums. The sum of an AK series is defined via any bijection f : ℕ → X, and Theorems 3.1.3 and 3.1.6 prove commutativity and associativity of this sum, even when X is infinite. By introducing an equivalence relation ∼ (bijection‑preserving equality of terms), Klazar shows that the quotient class T = S/∼ forms a semiring under natural addition and multiplication of series. This framework generalizes the usual notion of series and provides a clean algebraic setting for many “sums over sets” that appear in measure theory and probability.

Chapter 4 treats limits of real‑valued functions. After the standard definitions of one‑sided and two‑sided limits, the text presents strengthened versions of the arithmetic of limits (Theorem 4.4.1) and a refined squeeze theorem (Theorem 4.4.13). Section 4.6 is devoted to limits of inverse functions, culminating in a series of theorems (4.6.4, 4.6.9, 4.6.12, 4.6.17) that give precise conditions under which the limit of an inverse equals the inverse of a limit—results not usually found in standard textbooks. Asymptotic notation is revisited in Section 4.7, where Klazar defines asymptoticity for binary relations on the space of real functions, a subtle generalization that can accommodate more exotic growth comparisons.

Chapter 5 introduces elementary functions in a fully rigorous way. After defining basic exponentials, logarithms, and trigonometric functions, the author proves the fundamental identities e^{x+y}=e^{x}e^{y} (Theorem 5.1.4) and Euler’s formula e^{it}=cos t+i sin t (Theorem 5.1.30). Polynomials and rational functions are given canonical forms (Theorems 5.3.5 and 5.3.20), and the whole family is placed under the umbrella term “elementary functions” (Definition 5.2.7).

Chapter 6 focuses on continuity. The book supplies a novel proof of Blumberg’s theorem (Theorem 6.1.15) using a method labeled “MA 1+”, and presents Sierpiński’s theorem (Theorem 6.2.4) proved within ZF, thereby minimizing reliance on the axiom of choice. The cardinality result (Theorem 6.3.5) shows a bijection between the set of continuous functions ℝ→ℝ and ℝ itself, a striking illustration of the “size” of the continuous function space. Uniform continuity is treated in depth; Theorem 6.6.10 proves that any uniformly continuous function extends uniquely to the closure of its domain, and Theorem 6.6.11 leverages this to generalize the minimax theorem to uniformly continuous functions on bounded domains.

Chapter 7 re‑examines differentiation. Klazar defines the derivative at any limit point of the domain, not only at interior points, and treats differentiation as a unary operator on ℝ. The usual necessary condition for extrema (Theorem 7.1.9) and a concrete example of a discontinuous derivative (Theorem 7.1.29) are given. Tangents are formalized as limits of secants (Definition 7.2.10, Theorem 7.2.15), and the interaction of differentiation with arithmetic, composition, and inversion is explored in Sections 7.3–7.4. A noteworthy result (Theorem 7.6.5) shows that the subclass of “simple elementary functions” is closed under differentiation, while the broader question for all elementary functions is left as an open problem (Problem 7.6.1).

Chapter 8 presents mean‑value theorems. After restating Rolle’s, Lagrange’s, and Cauchy’s theorems, Klazar extends them to more general domains (Theorems 8.1.5 and 8.1.8). Applications include a proof that the sequence (log n) is not P‑recurrent (Section 8.2), and two independent constructions of transcendental numbers using Cantor’s and Liouville’s methods (Sections 8.3 and 8.4). Section 8.5 revisits monotonicity and L’Hospital’s rule, while Section 8.6 discusses higher‑order derivatives and their relationship to convexity/concavity (Theorem 8.6.11). Finally, Section 8.7 offers a 13‑step algorithm for sketching graphs of functions, illustrated with sgn x, tan x, and arcsin(2x/(1+x²)).

Chapters 9 and 10 develop Taylor polynomials, series, and real‑analytic functions. The text gives a systematic algebra of Taylor polynomials (Section 9.3) and treats analytic functions via convergent power series (Section 10.3). Additional topics include asymptotics of ordered partitions (10.4), Arnold’s limits (10.5), and inverses of Taylor series (10.6).

Chapters 11–13 introduce three integral concepts in succession: Newton’s integral, the Riemann integral, and the Henstock–Kurzweil integral, each with its fundamental theorem and illustrative applications. Chapter 14 surveys applications of integrals to geometry, probability, and physics.

Appendix A collects auxiliary material: logical and set‑theoretic notation, a brief overview of ZFC with classes, a philosophical note on proof verification, complex numbers, and metric spaces. Appendix B provides complete solutions to every exercise, making the book a self‑contained learning environment.

Overall, Klazar’s manuscript stands out for its blend of rigorous classical material with original contributions—most notably the AK series framework, the refined treatment of limits of inverse functions, and the choice‑free proofs of Blumberg and Sierpiński theorems. The structure (summary → sections → passages → exercises → solutions) encourages active learning, while the frequent “highlights” sections point out where the author’s approach diverges from standard textbooks. For instructors seeking a modern, proof‑oriented text that also offers fresh research‑level insights, “Univariate Real Analysis” is a valuable addition to the literature.