Let $\sigma:\Sigma\rightarrow \Sigma$ be the shift map on the space $\Sigma=\mathbb{N}^{\mathbb{N}}_+$ of sequences $a=(a_n)_{n\in \mathbb{N}}$ with $a_i \in \mathbb{N}_+=\mathbb{N}\backslash \left\lbrace 0 \right\rbrace$ positive integers. Let $X=[0,1]\backslash\mathbb{Q}$ the set of irrational numbers in $[0,1]$ and consider the map $\psi:\Sigma\rightarrow X$ given by $$\psi(a)=[a_0,a_1,\ldots]=\dfrac{1}{a_0+\dfrac{1}{a_1+\ldots}}.$$
The goal in this problem is to prove that $\psi$ is a conjugacy between the Gauss map $G : X\rightarrow X$ and the shift $\sigma:\Sigma\rightarrow \Sigma$.
Im in trouble in the part when I want to prove that $\psi$ and its inverse $\psi^{-1}$ are continuous. Can someone help please. Thank you