Pathological Periodicity

An exploration of the periodic functions \(f:\mathbb{R}\to\mathbb{R}\) that admit what may be called “incommensurable decompositions”, the pathologies that arise when the axiom of choice is assumed, and related matters in real analysis and set theory.

Posts are listed in reading order.

Pathological Periodicity, Part I: Relearning the Basics

Let me begin with a more narrative introduction to this series, which will be followed by a more technical one in the next post.

Pathological Periodicity, Part II: The Cage of Continuity

In the previous post, we motivated a series of questions surrounding the sums of periodic functions and the existence of functions with incommensurable periods. Now, we are…