Continuity of the continued fraction mapping revisited

TL;DR

This paper revisits the continuity properties of the continued fraction mapping on the unit interval, analyzing its behavior at both rational and irrational points to deepen understanding of its topological characteristics.

Contribution

It provides a detailed examination of the continuity of the continued fraction mapping at rational and irrational points, clarifying its topological properties.

Findings

The continued fraction mapping is a homeomorphism on irrationals.

Continuity at rational points is carefully characterized.

Insights into the topological structure of the continued fraction space.

Abstract

The continued fraction mapping maps a number in the interval $[0,1)$ to the sequence of its partial quotients. When restricted to the set of irrationals, which is a subspace of the Euclidean space $\mathbb{R}$, the continued fraction mapping is a homeomorphism onto the product space $\mathbb{N}^{\mathbb{N}}$, where $\mathbb{N}$ is a discrete space. In this short note, we examine the continuity of the continued fraction mapping, addressing both irrational and rational points of the unit interval.