Riemann's function with infinite points of discontinuity
- Juan Carlos Ponce Campuzano
Riemann's example of an integrable function with infinite points of discontinuity. Of course, we cannot really draw it but at least we can have an idea of how it looks with a sufficient number of elements of the serie: . Click on the Animation bottom. The function is defined as follows: , which is discontinuous on a dense subset of the real numbers. Nevertheless, is integrable.
You can check Riemann's example in: Riemann, Bernhard, 1902. Gesammelte Mathematische Werke und Wissenschaftlicher Nachlass, 2nd ed., and Nachträge. Leipzig, B. G. Teubner. p.228 Link:https://archive.org/details/bernardrgesamm00riemrich