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$\newcommand{\RR}{\mathbb{R}}\newcommand{\QQ}{\mathbb{Q}}$One can show that there are uncountably many continuous function $f:\RR\to\RR$ with the property that $f(q)\in\QQ$ for $q\in\QQ$. It's also not hard to show that there are still uncountable many such functions which are $C^\infty$. My questions is:

Are there uncountably many analytic functions $f:\RR\to\RR$ for which $f(q)\in\QQ$ for every $q\in\QQ$.

Dirk
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1 Answers1

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Yes. See this beautiful answer by Makholm and note that there are uncountably many enumerations of the rationals (there are plenty of parameters open to being varied here, but I guess this is the first that catches the eye in the answer).

Fimpellizzeri
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  • Also, note this link in the comments of the linked question, which also answers the question, but in more general terms. Makholm's answer bakes even more properties into the analytic function (behavior at infinity). – Fimpellizzeri Nov 26 '19 at 12:29