Continuous bijection from $\mathbb{R}^{2} \to \mathbb{R}$ Can anyone give an example of a continuous bijection from $\mathbb{R}^{2} \to \mathbb{R}$

Suppose $f:\mathbb R^2\to\mathbb R$ is a function.  If $f$ is injective then there exist points $a<b<c$ in the range of $f$.  Let $P=f^{-1}(b)$.  Then $\mathbb R^2\setminus\{P\}$ is the union of the disjoint nonempty sets $f^{-1}(-\infty,b)$ and $f^{-1}(b,\infty)$.  If $f$ is also continuous, then these sets are open.  This violates connectedness of $\mathbb R^2\setminus\{P\}$.
