Discontinuity and Involutions on Countable Sets

TL;DR

This paper constructs a function on infinite rational subsets with specific discontinuity points, demonstrating a novel way to control involutions and discontinuities on countable sets.

Contribution

It introduces a method to explicitly construct involutive functions on countable sets with prescribed discontinuity sets, expanding understanding of function behavior on rationals.

Findings

Existence of involutive functions with prescribed discontinuities on rationals

Construction method for functions with no isolated points in the discontinuity set

Advances in understanding discontinuities on countable dense subsets

Abstract

For any infinite subset $X$ of the rationals and a subset $F \subseteq X$ which has no isolated points in $X$ we construct a function $f: X \to X$ such that $f(f(x))=x$ for each $x\in X$ and $F $ is the set of discontinuity points of $f$.