TITLE: Does there exist an injective function $f:\mathbb R^2 \to \mathbb R$ such that $f$ is continuous in one of the variables $x$ or $y$?
QUESTION [5 upvotes]: Does there exist an injective function $f:\mathbb R^2 \to \mathbb R$ such that $f$ is continuous in one of the variables $x$ or $y$ ? I only know that such an injection cannot be continuous in each real variable $x$ and $y$ . Please help . Thanks in advance

REPLY [2 votes]: No. Here is an outline:
Suppose $g:\mathbb{R} \to \mathbb{R}$ is continuous and injective, then $g(\mathbb{R})$ is a non trivial interval.
Suppose $I_\alpha$ is an uncountable collection of non trivial intervals. Then at least two of the intervals overlap.
Now consider the uncountable non trivial collection $\{f(\mathbb{R},y) \}_y$