Arithmetic Rigidity of Analytic Functions and Mahler's Problem on Liouville Numbers

Let $\mathscr L$ denote the set of Liouville numbers. We prove a local arithmetic rigidity theorem for real-analytic functions. If $U\subset\mathbb R$ is an open interval and $f:U\to\mathbb R$ is real-analytic with $f(\mathscr L\cap U)\subseteq\mathscr L$, then $f$ is the restriction to $U$ of a rat