  Suppose that $F(x)\in\mathbb{Z}[[x]]$ is a Mahler function and that $1/b$ is
in the radius of convergence of $F(x)$. In this paper, we consider the
approximation of $F(1/b)$ by algebraic numbers. In particular, we prove that
$F(1/b)$ cannot be a Liouville number. If $F(x)$ is also regular, we show that
$F(1/b)$ is either rational or transcendental, and in the latter case that
$F(1/b)$ is an $S$-number or a $T$-number.
