Nonexistence of an injective $C^1$ map between $\mathbb R^2$ and $\mathbb R$

real-analysis general-topology

Solution 1:

There is no such map.

If $f\colon\mathbb R^2\to\mathbb R$ is continuous then its image is connected, that is an interval in $\mathbb R$. Note that this is a non-degenerate interval since the function is injective.

However if you remove any point from $\mathbb R^2$ it remains connected, however if we remove a point whose image is in the interior of the interval then the image cannot be still connected if the function is injective.

Solution 2:

For what its worth, there isn't even any continuous injection from $\mathbb{R}^m$ into $\mathbb{R}$ for $m > 1$

The proof follows the exact same argument as Asaf's does.

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