Continuity of a mapping $C\to C^2$, $C$ being the Cantor set I will denote the Cantor set as $C$. We have proved earlier that every $x\in C$ can be uniquely written in a ternary representation $x=0.a_1a_2a_3...$ where all the $a_i \in \{0,2\}$.
Now we consider the function $$f:C\to C^2 \\ 0.a_1a_2a_3... \mapsto \left( 0.a_1a_3a_5... , 0.a_2a_4a_6... \right)$$
I would like to show that $f$ is continuous.
I appreciate any advice/hints.

Let $f=(f_1,f_2)$, with $f_1,f_2: C\to C$.
Clearly, if 
$\lvert x-y\rvert<\delta$, and $\delta<\dfrac{2}{3^{2n+1}}$, for some $n$, then the first $2n+1$ digits in the ternary expansions of $x$ and $y$ agree, and hence so do the first $n$ ternary digits of $f_i(x)$ and $f_i(y)$, $i=1,2$, and thus
$$
\lvert\, f_1(x)-f_1(y)\rvert,\,\lvert\, f_2(x)-f_2(y)\rvert<\dfrac{2}{3^{n}}.
$$
Hence, for every $\varepsilon>0$, there exists a $\delta=\dfrac{2}{3^{k}}$, where $$
k=2\lfloor\log_3 (\varepsilon/2)\rfloor+1,
$$
such that...
