Let $C:\mathbb{R} \to \mathbb{R}$ be the Conway Base-13 function.
My question: It seems obvious to me that $f:\mathbb{R} \to \mathbb{R}^2$ defined by $f(x) = (C(x),C(x^2))$ should take every non-empty open interval to all of $\mathbb{R}^2$. Does it suffice as proof to say that because $C$ takes every open interval to all of $\mathbb{R}$, that the same would be true for $f$? I am not sure how to formalize this.
The actual question asked was to find a function from $\mathbb{R} \mapsto \mathbb{R^2}$ that is a surjection, but nowhere continuous. I can come up with other examples, but the Conway Base-13 function is one of my favorite nowhere continuous functions, and I am not even sure I can come up with a proof that it satisfies surjectivity overall. Still, I was hoping to show that the original function's property of taking every interval to the entire set still held.