Let $n$ be a positive integer. Is it possible to color every point in the plane in one of $n$ colors so that every (nondegenerate) circle contains points of every color?
If we can do the coloring so that every arbitrarily small interval of $x$-coordinate contains all the colors, we would be done, since a circle must contain all points of some interval of $x$-coordinate. This reduces the problem to $1$ dimension. But how can we do this coloring?
Your second paragraph proves that we can do this with continuum many colors. The two answers to this question and the ones here describe functions that take all real values in any interval of $x$.
Let $p_i$ be the $i$th prime; and define $A_i = \{r \in \mathbb Q : \exists a,b \in \mathbb Z $ with $gcd(a,p_i)=1$, $b>0$, and $ r = \frac{a}{(p_i)^b}\}$. Each $A_i$ is dense in $\mathbb R$ and they are mutually disjoint; so color each point $(x,y) \in \mathbb R^2$ as color $i$ if $y \in A_i$ (and color 1 if $y$ is not in any $A_i$).
