  In 1906, Maillet proved that given a non-constant rational function $f$, with
rational coefficients, if $\xi$ is a Liouville number, then so is $f(\xi)$.
Motivated by this fact, in 1984, Mahler raised the question about the existence
of transcendental entire functions with this property. In this work, we provide
an uncountable subset of Liouville numbers for which there exists a
transcendental entire function taking this set into the set of the Liouville
numbers.
