Elementary Real Analysis (Klazar)

martin-klazar's real-analysis book, arxiv:2508.19405, v14 of 16 July 2026, 400 pages, CC BY 4.0. It grew out of his lecture notes for Mathematical Analysis 1, taught for several years in the School of Computer Science at MFF UK in Prague, and it is openly unfinished: the title page says "Preliminary version of July 17, 2026" and the colophon dates the work "August 2024 to ??? 2026."

It is dedicated to jiri-matousek, Klazar's colleague, who "was preparing lectures for the analysis course in the school year 2014/15, but illness did not allow him to deliver them."

Klazar states the motive without decoration: "Why write another text? … Simply put, I want to write it in a better way." He then lists sixteen existing texts he is writing against, and adds — this is the whole sentence — "(Many authors of the cited works were clearly motivated similarly.)"

Four chapters do not exist

Chapters 11 to 14 — Newton integral, Riemann integral, Henstock–Kurzweil integral, Applications of integrals — are title pages and nothing else, one page each. A four-hundred-page book called Elementary Real Analysis currently contains no integration at all.

That is not concealed. The table of contents marks "(c … means a relatively complete section)" and carries the marker on Chapter 1 throughout, on Chapter 7's first four sections, and on parts of Appendix A — a completeness index printed in the front matter, which no other text in this corpus offers. Compare carroll-geometric-function-theory's §1.4 "What Is Not in This Book": both authors tell you where the book stops, one about scope and one about progress.

So the shelf position is specific. trench-real-analysis is the spoke's complete two-term analysis text, integrals and several variables included. This one goes deeper into the ground floor and has not reached the first integral.

The ground floor, rebuilt

Chapter 1 is the reason to hold the book. It develops five numeric domains — ℕ₀, ℤ, ℚ, ℝ and the extended reals ℝ* — and gives each an algebraic characterization theorem, of which Klazar says "only the one for ℝ is well known, and the last one for ℝ* is completely new." Up to isomorphism: ℕ₀ is the unique simple ordered semiring, ℤ the unique simple ordered ring, ℚ the unique simple ordered field, ℝ the unique complete ordered field, and ℝ* the limit closure of the linear order ℝ.

Two consequences worth naming:

Induction is a theorem, not an axiom. ℕ₀ is built as the set ω inside ZFC via the axiom of infinity, and "the principle of induction for natural numbers is, in our approach, a theorem following from more fundamental set-theoretic axioms."
The extended reals get a structure, not a convention. Most texts introduce ±∞ as notation with rules attached; here ℝ* is characterized and its partial operations are described by theorem.

ℝ itself arrives by Dedekind cuts, the same route baggett-analysis-single-variable takes in an appendix.

A combinatorialist's analysis book

Klazar works in MFF UK's applied-mathematics department, and the examples give him away. Fekete's lemma in combinatorics is a section of the sequences chapter. The series chapter opens with set series and generalizes Pólya's theorem before reaching the classical tests. Later sections cover Sierpiński's theorem, the cardinality of the set of continuous functions, asymptotics of ordered partitions, and Arnol'd's limits. The plan includes the Henstock–Kurzweil integral, which is unusual in a first course.

The book also argues with its ancestors rather than citing them. One passage takes Cauchy's Cours d'analyse (1821), quotes Theorem I of §2.3 — if f(x+1) − f(x) converges to k then f(x)/x converges to the same k — and presents counterexamples to it, as an exercise in functional limits. Structure throughout: chapter summary → sections → passages, and every passage carries at least one exercise, with solutions and hints in Appendix B (pages 335–372).

The fifth assumer, and the strongest case yet

This spoke's sharpest open edge asks for a text that argues for the ε-δ foundation rather than simply adopting it — something that considered non-standard analysis and declined it in print, against keisler-elementary-calculus, which builds the whole calculus sequence on infinitesimals and has an experiment behind its pedagogy.

Klazar does not close it. Searched in full: no Keisler, no Robinson, no hyperreals, no "non-standard analysis." The word infinitesimal occurs three times — twice as the thing an Archimedean field is defined not to contain, once describing Cauchy's 1821 book as "a seminal textbook in infinitesimal calculus." ("Non-standard" appears once, unrelated, about his own definition of Taylor polynomials.)

He is the fifth text here to assume the standard development, and he is the worst case for the merit reading, worse than baggett-analysis-single-variable was. Baggett built ℝ from cuts in an appendix without naming the alternative. Klazar re-derives the entire numeric hierarchy from ZFC axioms, proves induction rather than assuming it, adds a new characterization theorem, writes an appendix section titled "How do we know that a theorem is true?", and lists sixteen rival texts he means to improve on — an author auditing the foundations of the subject from the bottom, in public, across fourteen versions. If anyone were going to weigh the infinitesimal route and reject it on the record, it is this book. It does not mention that the route exists.

The edge stands, and it now has a better justification than "nobody got round to it."

Licence and tier

CC BY 4.0 — attribution only. That is more permissive than trench-real-analysis's CC BY-NC-SA and herzog-understanding-statistics's CC BY-NC, and short of grinberg-graph-theory's CC0. The spoke's second arXiv source, and its second-most-open licence.

T1 — the author's own text, read from the PDF. freshness: volatile for a live-drafted book on its fourteenth version: the chapter list will change, and the four empty chapters are the specific thing to re-check.

trench-real-analysis · baggett-analysis-single-variable · keisler-elementary-calculus · grinberg-graph-theory · carroll-geometric-function-theory · martin-klazar · jiri-matousek · charles-university · arxiv · synthesis