## Real analytic function, injective, non surjective and preserving the rationals ?

I'd like to prove the non-existence of a real analytic function, injective, non-surjective that sends rationals to rationals. Is it a classical result ? If not, any hints on how to prove it ? Thanks in advance for you help.

-
Does the function sends all rationals to all rationals? – Math-player Sep 24 2011 at 22:48
By continuity, it would be surjective, if it surjective on a dense subset! So not all rationals are in the image. – Marc Palm Sep 24 2011 at 22:58
I don't know the answer, but I do know that if you reduce real-analytic to continuous, there is a function (pick any order isomorphism from Q to the negative rationals, and extend it to R) – Richard Rast Sep 25 2011 at 0:23
If you drop analytic, take $x / (1 + |x|)$ – Will Jagy Sep 25 2011 at 1:36
According to the following paper, the statement in the question is not true. Given any two enumerable and dense sets in open intervals of the reals, there is a real analytic function giving a bijection between them: Analytic Transformations of Everywhere Dense Point Sets, Philip Franklin, Transactions of the American Mathematical Society, Vol. 27, No. 1 (Jan., 1925), pp. 91-100. jstor.org/stable/1989166 – George Lowther Sep 25 2011 at 2:30