Ejemplo de función que no es Riemann-Integrable

Ejemplo de función que no es Riemann-Integrable

by Efraín Vega Landa

Formal & Physical Sciences Mathematics PBKCalculus and mathematical analysisPBKACalculus
🎙 Efraín Vega Landa 👥 117K 📅 August 26, 2026 ⏱ 82 min 👁 63 📄 tutorial 🧭 2026-08-26
Available in: English (current) Français

Keywords

Riemann sumpartitiontagged partitionDirichlet functionintegrability

Summary

The video is a lecture by Efraín Vega Landa, part of a calculus course, aiming to motivate the definition of the Riemann integral and to present a function that is not Riemann-integrable. The instructor begins by recalling the concept of Riemann sums, where the interval [a,b] is partitioned into subintervals, and a sample point is chosen in each subinterval to form rectangles approximating the area under the curve. He emphasizes two generalizations: the choice of sample points (not necessarily endpoints) and the possibility of non-uniform partitions. The formal definition of the Riemann integral is then given as the limit of Riemann sums as the mesh of the partition tends to zero, independent of the choice of tags. The instructor notes that continuous functions are always integrable, but then introduces the Dirichlet function, which is 1 on rationals and 0 on irrationals on [0,1]. He explains that its graph is dense in the unit square, and that any Riemann sum can be made to approach either 0 or 1 depending on the choice of tags, so the limit does not exist. Thus, the function is not Riemann-integrable. The lecture concludes with a discussion of the implications and a preview of the Lebesgue integral as a more powerful alternative.

207 words

Critical Evaluation

Value of the Information & Strength of the Argument

The video provides a solid, rigorous explanation of the Riemann integral and the Dirichlet function as a counterexample. The argumentation is clear and logical: it builds from the intuitive idea of approximating area with rectangles, formalizes the Riemann sum, and then demonstrates the failure of integrability for the Dirichlet function by showing that the limit of Riemann sums depends on the choice of sample points. The instructor uses a pedagogical approach, asking questions and engaging with the audience, which helps clarify the concepts. The value lies in its didactic presentation of a fundamental example in real analysis, making it accessible to students.

Scientific Rigor, Source Quality, Title Accuracy

The mathematical content is rigorous and accurate. The instructor correctly defines Riemann sums, partitions, and the integral, and the Dirichlet function is presented as a standard example of a non-Riemann-integrable function. The title accurately reflects the content. No external sources are cited, but the lecture is self-contained and relies on standard definitions from calculus. The video is a classroom recording, so the production quality is informal, but the mathematical exposition is precise.

189 words

Title / Content Match

The title accurately describes the content: the video presents a function that is not Riemann-integrable, namely the Dirichlet function, and explains why it fails the definition.

Quality & Reliability

8/10

The video is a rigorous, formal lecture on Riemann integration, presenting the definition of the Riemann sum, the role of partitions and tags, and the Dirichlet function as a non-integrable example. The mathematical content is correct and well-structured, though it is a classroom recording with some informal digressions.

Key Moments

Markers derived by PSI from the transcript: the creator did not define chapters.

Contribution & Novelties

The video provides a clear pedagogical explanation of a classic counterexample in real analysis, the Dirichlet function, and its non-Riemann-integrability. It emphasizes the importance of the choice of sample points and the mesh of the partition, which are often glossed over in introductory calculus. The lecture also hints at the need for a more powerful integration theory, such as the Lebesgue integral.

Pour aller plus loin :

  • Riemann integral — Provides the formal definition and properties of the Riemann integral.
  • Dirichlet function — Detailed discussion of the function and its properties.
  • Lebesgue integral — Introduces a more general integration theory that can handle functions like the Dirichlet function.

108 words

Radar Profile

The radar profile shows high scores in quality, technical level, and reliability, reflecting the rigorous mathematical content and clear exposition. The quantity of information is also high, as the lecture covers definitions, examples, and implications. The overall profile indicates a solid educational resource.

Reliability 9/10