There must be a map from $\mathbb R^2$ to $\mathbb R$, since they are the same cardinality. But is there a construction for a continuous map from $\mathbb R^2$ to $\mathbb R$?

I guess what I mean is a map $M : \mathbb R^2 \to \mathbb R$ such that as $x$ approaches $x'$, and $y$ approaches $y'$, $M(x, y)$ approaches $M(x', y')$.

If the above doesn't make sense, my apologies, and I hope you are able to figure out what I meant.