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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 $f=id$?

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    Looks like the answer to my question answers yours: http://mathoverflow.net/questions/46854/continuity-in-terms-of-lines/46860#468602010-11-21

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(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

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    @Peter: you were just 1 rep short! I think now you can. (I don't know the proof of the Fund. Thm myself, so I didn't feel comfortable elaborating). Cheers2010-11-21