Is there an injective continuous map $\mathbb{R}^2 \rightarrow \mathbb{R}$?

Question

It is commonly known fact that there exists a continuous surjective map $\mathbb{R} \rightarrow \mathbb{R}^2$. So it bids to ask:

Is there an injective continuous map $\mathbb{R}^2 \rightarrow \mathbb{R}$?

Answer 1

No. If so by connectivity the image would be an interval. Removing any $3$ points in the image will disconnect it, however, removing the corresponding $3$ points in the domain will not disconnect $\mathbb{R}^2$. Then we have a continuous map from a connected set onto a disconnected set.

This also shows there isn't a continuous finite to $1$ map from $\mathbb{R}^2$ to $\mathbb{R}$. Actually it shows there isn't a continuous countable to $1$ map from $\mathbb{R}^n$ to $\mathbb{R}$ because $\mathbb{R}^n$ with a countable set removed is path connected.