  If $\varphi$ is an analytic function from the unit disk $\mathbb{D}$ to
itself, and $\varphi$ is not a conformal automorphism, we denote by
$\lambda_{\varphi}$ its Denjoy-Wolff point, that is, the limit of the iterates
$\varphi(\varphi(\cdots\varphi(0)\cdots))$. A result of Heins shows that, given
a sequence $(\varphi_{n})_{n\in\mathbb{N}}$ of such analytic functions that
convergence pointwise to $\varphi$, it follows that
$\lim_{n\to\infty}\lambda_{\varphi_{n}}=\lambda_{\varphi}$. This allows us to
improve results about the contnuous extensions of the subordination functions
that arise in the study of free convolutions. We also offer an alternate proof
of the result of Heins.
