On 10/02/2016 05:50, Dan Asimov wrote:
Does there exist a real analytic function
f: R —> R
that takes rationals to rationals, but is not a rational function?
http://mathoverflow.net/questions/48910/smooth-functions-for-which-fx-is-rat... has (for its top-rated answer) a pretty straightforward construction giving you C^\infty functions that do this while simultaneously looking "approximately" more or less however you like. The second answer links to http://mathoverflow.net/questions/42460/is-a-real-power-series-that-maps-rat... whose second answer appears to give a construction for an entire (complex) function mapping rationals to rationals. Of course it also maps reals to reals, so its restriction to R is a real-analytic function mapping rationals to rationals. I'm not sure how easy it is to make sure that the resulting function doesn't happen to be a rational function (maybe it's trivial and I'm just not seeing how?). -- g