Continuous surjection from $[0,1] \rightarrow [0,1]^2$ I am currently exercising on the Cantor set $C$, and I know the following facts from previous examples. The Cantor set is constructed in this case recursively, such that $C_{n+1} := (\frac{1}{3} C_n) \cup (\frac{1}{3}C_n + \frac{2}{3})$ and $C := \bigcap_{n\in\mathbb{N}_0} C_n$. The function $g: C \rightarrow [0,1]$, $\sum_{k=1}^{\infty} (a_k+1) 3^{-k} \mapsto \sum_{k=1}^{\infty} (a_k+1) 2^{-k-1}$ is a surjective, monotone increasing function. By Urysohn's lemma and Tietze extension we can construct a continuous extension of $g$, which I'll denote by $G$ on $[0,1]$. From this, I have proven, that there must exist a continuous surjection $h: C \mapsto [0,1]^2$. I could also extend $h$, and then I'll get the function
$$H:[0,1] \mapsto [0,1]^2, \quad t \mapsto \frac{1}{b-a} (h(b)(t-a) + h(a)(b-t)).$$
$(a,b) \subset [0,1]$ should be an interval removed by the construction of $C$. I would like to proof, that this function is a continuous surjection, I actually should be able to apply the previous proofs as long as I can show that it is continuous on $\frac{2}{3}$.

You know that there exists a continuous surjection $h: C \to I^2$. Now you can again apply the Tietze extension theorem:
Let $h_1, h_2 : C \to I$ denote the coordinate functions of $h$, i.e. $h(x) = (h_1(x),h_2(x))$. Both have continuous extensions $H_i : I \to I$. Take $H(x) = (H_1(x),H_2(x))$.
