Existence of an injective continuous function $\Bbb R^2\to\Bbb R$? Let's say $f(x,y)$ is a continuous function. $x$ and $y$ can be any real numbers. Can this function have one unique value for any two different pairs of variables? In other words can $f(a,b) \neq f(c,d)$ for any $a$, $b$, $c$, and $d$ such that $a \neq c$ or $b \neq d$? I don't think there can at least not if the range of $f$ is within the real numbers. Could someone please offer a more formal proof of this or at least start me off in the right direction.

Assume $f : \mathbb{R}^2 \to \mathbb{R}$ is continuous and injective. Then for each fixed $y$, the function $x \mapsto f(x,y)$ is monotonic. Its image is some interval, and in particular contains a rational number. None of these points can be re-used for some other $y$. So $y$ can't be drawn from an uncountable set, since the rationals are countable. But $\mathbb{R}$ is uncountable...
