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 $h$ be the mapping between the so-called Cantor cube $K = \{0,1\}^{\mathbb{N}}$ (in the product topology, where $\{0,1\}$ is discrete) and the Cantor set (as a subset of $[0,1]$) that sends a sequence $(x_i)_{i \ge 1} \in K$ to the point in $C$ with ternary expansion $0.a_1a_2\ldots$ where $a_i = 2x_i$ for all $i$.
Key to the proof is the fact that $h$ is a homeomorphism. Once you know that, your $f$ is just the index shuffling homeomorphism between $K$ and $K \times K$, which is easily shown to be continuous using the universal property of maps into product (continuous iff all the compositions with projections are). 
