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 R, the continued fraction mapping is a homeomorphism onto the product space NN, where 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.
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Min Woong Ahn (2024) studied this question.
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