Suppose that F ( x ) ∈ Z [ [ x ] ] is a Mahler function and that 1 / b is in the radius of convergence of F ( x ) for an integer b ⩾ 2 . 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, in addition, F ( x ) is 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 in Mahler's classification of real numbers.