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arxiv: 2508.19405 · v11 · submitted 2025-08-26 · 🧮 math.HO

Univariate Real Analysis

Martin Klazar This is my paper

Pith reviewed 2026-05-18 21:41 UTC · model grok-4.3

classification 🧮 math.HO
keywords univariate real analysisreal numberslimitsderivativesTaylor polynomialsRiemann integralHenstock-Kurzweil integralapplications of integrals
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The pith

This manuscript provides a complete preliminary treatment of univariate real analysis in 14 chapters plus appendices.

A machine-rendered reading of the paper's core claim, the machinery that carries it, and where it could break.

The paper organizes univariate real analysis into a sequence of chapters that starts with the real numbers and moves through limits, continuity, differentiation, and several approaches to integration. A sympathetic reader would value this because it assembles the core definitions, theorems, and proofs of the subject into one self-contained resource rather than scattering them across multiple texts. If the presentation holds, students and instructors obtain a clear path from axioms about real numbers to applications of integrals, including both classical and more general integration theories.

Core claim

The manuscript constitutes a preliminary version of a book on univariate real analysis organized into 14 chapters and 2 appendices that cover real numbers, limits of sequences, series, limits of functions, elementary functions, continuous functions, derivatives, mean value theorems, Taylor polynomials, real analytic functions, the Newton integral, the Riemann integral, the Henstock-Kurzweil integral, applications of integrals, auxiliary notions, and solutions to exercises.

What carries the argument

The ordered progression of chapters that builds from the properties of real numbers through increasingly general notions of integration.

If this is right

  • The text supplies a unified account that places the Henstock-Kurzweil integral alongside the Riemann and Newton integrals.
  • Chapters on Taylor polynomials and analytic functions link differentiation to local power-series behavior.
  • The applications chapter illustrates concrete uses of the integral concepts developed earlier.
  • Exercises together with their solutions support independent verification of the main results.

Where Pith is reading between the lines

These are editorial extensions of the paper, not claims the author makes directly.

  • Placing three distinct integral theories in consecutive chapters makes direct comparisons of their strengths and limitations easier than in most textbooks.
  • The structure could be adapted for a course that emphasizes the historical development of integration concepts.
  • Including solutions to exercises reduces the barrier for self-study of the more technical later chapters.

Load-bearing premise

The detailed content of the chapters presents standard definitions, theorems, and proofs of real analysis without introducing errors or significant pedagogical gaps.

What would settle it

An incorrect statement or flawed proof appearing in any chapter, for example a mistaken characterization of the Henstock-Kurzweil integral or its relation to the Riemann integral, would show the treatment is not yet reliable.

read the original abstract

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

Editorial analysis

A structured set of objections, weighed in public.

Desk editor's note, referee report, simulated authors' rebuttal, and a circularity audit. Tearing a paper down is the easy half of reading it; the pith above is the substance, this is the friction.

Referee Report

0 major / 2 minor

Summary. The manuscript is a preliminary version of a book-length exposition on univariate real analysis, organized into 14 chapters and 2 appendices. The chapters cover the construction and properties of real numbers, limits of sequences and functions, infinite series, elementary functions, continuity, differentiation, mean value theorems, Taylor polynomials and series, real analytic functions, the Newton integral, the Riemann integral, the Henstock-Kurzweil integral, and applications of integration, with appendices on auxiliary notation and solutions to exercises.

Significance. If the detailed content accurately presents standard definitions, theorems, and proofs without introducing errors, the work offers a comprehensive, self-contained treatment suitable for advanced undergraduates or beginning graduate students. The inclusion of solutions to exercises in Appendix B is a clear pedagogical strength that supports independent study. Coverage of the Henstock-Kurzweil integral alongside classical integrals provides a modern perspective on integration theory that is not always present in introductory texts.

minor comments (2)
  1. [Front matter] The table of contents and abstract are consistent in describing the chapter structure, but the manuscript would benefit from an explicit preface or introduction section that states the intended audience, prerequisites, and any pedagogical innovations relative to existing texts such as Rudin or Apostol.
  2. [Appendices] In a book-length work, the absence of an index or detailed subject index in the appendices limits quick reference; adding one would improve usability.

Simulated Author's Rebuttal

0 responses · 0 unresolved

We thank the referee for the positive and accurate summary of our preliminary manuscript on univariate real analysis. We appreciate the acknowledgment of its comprehensive coverage across the 14 chapters and 2 appendices, its suitability for advanced undergraduates or beginning graduate students, the pedagogical value of providing solutions to exercises, and the modern perspective afforded by including the Henstock-Kurzweil integral alongside classical approaches. We will implement minor revisions to the preliminary version as recommended.

Circularity Check

0 steps flagged

No significant circularity in standard exposition

full rationale

The manuscript is a preliminary book-length exposition of standard univariate real analysis topics, with chapters covering real numbers through Henstock-Kurzweil integration and applications. It presents known definitions, theorems, and proofs without novel derivations, predictions, or self-referential claims. The central claim is descriptive (organization into listed chapters matching the abstract), with no fitted inputs, self-citations as load-bearing premises, or reductions of results to their own inputs by construction. The work is self-contained against external benchmarks of standard real analysis and exhibits no circularity.

Axiom & Free-Parameter Ledger

0 free parameters · 0 axioms · 0 invented entities

No free parameters, new axioms, or invented entities are introduced; the work relies entirely on the established body of real analysis.

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