Convergence of Teichm\"uller deformations in the universal Teichm\"uller space

Let $\varphi :\mathbb{D}\to\mathbb{C}$ be an integrable holomorphic function on the unit disk $\mathbb{D}$ and $D_{\varphi}:\mathbb{D}\to T(\mathbb{D})$ the Teichm\"uller disk in the universal Teichm\"uller space $T(\mathbb{D})$. For a positive $t$ it is known that $D_{\varphi}(t)\to [\mu_{\varphi}]\in PML_b(\mathbb{D})$ as $t\to 1$, where $\mu_{\varphi}$ is a bounded measured lamination representing a point on the Thurston boundary of $T(\mathbb{D})$. We extend this result by showing that $D_{\varphi}\colon \mathbb{D}\to T(\mathbb{D})$ extends as a continuous map from the closed disk $\overline{\mathbb{D}}$ to the Thurston bordification. In addition, we prove that the rate of convergence of $D_{\varphi}(\lambda )$ when $\lambda\to e^{i\theta}$ is independent of the type of the approach to $e^{i\theta}\in\partial\mathbb{D}$.