Critical values of inner functions

Let $\mathcal{J}$ be the space of inner functions of finite entropy endowed with the topology of stable convergence. We prove that an inner function $F\in \mathcal{J}$ possesses a radial limit (and in fact, a minimal fine limit) in the unit disk at $\sigma (F^{\mathrm{\prime }})$ a.e. point on the unit circle. We use this to show that the singular value measure $\nu (F)={\sum }_{c\in \text{crit }F}(1-\mathrm{\mid }c\mathrm{\mid })\cdot {\delta }_{F(c)}+F_{\ast }(\sigma (F^{\mathrm{\prime }}))$ varies continuously in $F$. Our analysis involves a surprising connection between Beurling-Carleson sets and angular derivatives.