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I am not sure how to do this map. I don't know how I can place in particular that element that is in R/Q to R while still being an bijection.

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Do you mean $\Bbb{R/Q}$ or $\Bbb{R\setminus Q}$? – Asaf Karagila Dec 6 '12 at 0:53
Hi, I meant R\Q. Thanks – Jmaff Dec 6 '12 at 0:56

You can just use straight lines between the points. Say you want to fix $0,1,2$ and send $\sqrt 2$ to $\frac 32$. Then you have have

$$f(x)=\begin {cases} x & x \le 1 \\\\ 1+\frac {.5}{\sqrt 2-1}(x-1) & 1 \lt x \le \sqrt 2 \\\\ \frac 32+\frac {.5}{2-\sqrt 2}(x-\sqrt 2) & \sqrt 2 \lt x \le 2 \\\\x & x \gt 2\end {cases}$$

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