TITLE: Is a homeomorphism which maps lines to lines (and fixes zero) necessarily linear?
QUESTION [8 upvotes]: We know the homeomorphism $f: \mathbb{R}^n \rightarrow \mathbb{R}^n$ maps straight lines to straight lines and the zero vector to the zero vector. Is it Linear?? If so, how can we prove it?

REPLY [2 votes]: Are you asuming $n>1$? If not a good counter-example could be the function $\phi(x)=x^3$ ( $x\in\mathbb{R}$) . It's a homeomorphism, maps $\mathbb{R}$ to $\mathbb{R}$ (the only line in $\mathbb{R}$) and obviously it maps $0$ to $0$