Almost identical map
Let $f: \mathbb{R}^2 \rightarrow \mathbb{R}^2$ be bijective map with following properties:
1) $f|_{\mathbb{Q}^2}=id$;
2) Image of any line under map $f$ is again a line.
Is it right that $f=id$?
Solution 1:
(Posting this as CW so Alex can accept the answer.)
Trutheality re-asked the question on MathOverflow, and it turns out the answer is given by what is known as the "Fundamental Theorem of Affine Geometry". See https://mathoverflow.net/questions/46854/continuity-in-terms-of-lines