Continuous Maps From $\Bbb R^n$ to $\Bbb R$

Question

Is it possible to define a continuous 1-1 map from any open set $A \subset\mathbb{R}^n$ to $\mathbb{R}$?

I have trouble in giving a counter example here. I feel that it should not be possible but I may be wrong.

Answer 1

Not if $n > 1$. Suppose such a function existed. We may assume wlog $A$ is connected (else take a connected component). Then $f(A)$ is an interval. But removing a single interior point disconnects $f(A)$, while removing a single point of $A$ does not. And the continuous image of a connected set is connected.