Continuity of the continued fraction mapping revisited

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.