Continuous map between circles

Question

Let $C, C_1, C_2 \subset \mathbb{R}^2$ be circles such that $C_1 \cap C_2 = \emptyset$ and $C_1$ does not lie in the interior of the disk bounded by $C_2$ and similarly for $C_2$. Let $f: C \rightarrow C_1 \cup C_2$ be a map such that $f(x_1) \in C_1$ and $f(x_2) \in C_2$. Does it follow that $f$ is not continuous?

Answer 1

If $C_1$ and $C_2$ are disjoint, then $C_1$, $C_2$ are open in $C_1\cup C_2$ (why?), so if $f$ were continuous, $f^{-1}(C_1)$ and $f^{-1}(C_2)$ would be a disjoint cover of $C$ by nonempty open sets. But $C$ is connected, so that can't be.

The extra hypothesis, that each circle does not lie in the interior of the disk bounded by the other, seems superfluous.