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how should I define a one-to-one function from $(0,1)$ on to real numbers which maps rational numbers to rational numbers and irrational numbers to irrational numbers?

Asaf Karagila
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saleh
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  • How? You could start with something simpler, and maybe you can use some of the same techniques on your problem. Can you do it for $(0,1)\to (1, \infty)$ instead of $(0,1)\to (-\infty, \infty)$, for instance? – Arthur Jan 02 '20 at 14:33

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You can use the function $f(x)=1/x$ which maps $(0,1]$ to $[1,+\infty)$ and rescale it to map $(0,\frac12]$ to $[0,+\infty)$. You then use it again to map $[\frac12,1)$ to $(-\infty,0]$.

Luca Citi
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