If $f:{\mathbb{R}}^n\to{\mathbb{R}}^n$ $(n\ge2)$ is a bijection such that the image of every line is a line (continuity of $f$ *not assumed*), must $f$ be an affinity?


Assuming continuity would certainly suffice, even assuming that $f$ is order-preserving on each line. Is there a counterexample if we drop the assumption that $f$ is a bijection? Any references?



Line-preserving bijection of ${\mathbb{R}}^n$ onto itself
Yes $f$ must be an affinity – this is called the fundamental theorem of affine geometry and is found e.g. on page 52 of M. Berger's [*Geometry*](http://books.google.com/books?id=5W6cnfQegYcC&pg=PA52). (For other treatments and history, see [this question](https://mathoverflow.net/questions/191817/who-first-proved-the-fundamental-theorem-of-projective-geometry).)


