  We consider Mahler functions $f(z)$ which solve the functional equation $f(z)
= \frac{A(z)}{B(z)} f(z^d)$ where $\frac{A(z)}{B(z)}\in \mathbb{Q}(z)$ and
$d\ge 2$ is integer. We prove that for any integer $b$ with $|b|\ge 2$ either
$f(b)$ is rational or its irrationality exponent is rational. We also compute
the exact value of the irrationality exponent for $f(b)$ as soon as the
continued fraction for the corresponding Mahler function is known. This
improves the result of Bugeaud, Han, Wei and Yao where only an upper bound for
the irrationality exponent was provided.
