xtrm{if}\ (x_0,\omega_0,E_0)\in \Gamma_-.
\end{cases}$$ where $(x,\omega,E)\in G\times S\times I$ when taking the limits.

Define $$z_0=\begin{cases}
x_0-t(x_0,\omega_0)\omega_0,\ & \textrm{if}\ (x_0,\omega_0,E_0)\in G\times S\times I \\
x_0-\tau_+(x_0,\omega_0)\omega_0,\ & \textrm{if}\ (x_0,\omega_0,E_0)\in \Gamma_+ \\
x_0,\ & \textrm{if}\ (x_0,\omega_0,E_0)\in \Gamma_-.
\end{cases}$$ In each of the limits concerned, we have thus assumed that $(z_0,\omega_0,E_0)\in\Gamma_-$, which implies, using the fact that $\ol{G}$ is $C^1$-manifold with boundary and $\omega_0\cdot\nu(z_0)<0$, that there is an open finite cone $C\subset{\mathbb{R}}^3$ and $\lambda_0>0$ such that $z_0+\lambda_0 C\subset{\mathbb{R}}^3\backslash G$ and $-\omega_0\in C$. The claim then follows from Lemma \[le:esccont\]. Note in particular that when considering the limit $\lim t(x,\omega)=\tau_+(x_0,\omega_0)=\tau_-(y_0,\omega_0)$ one takes in the lemma $y_0=x_0-\tau_+(x_0,\omega_0)\omega_0$ and (hence) $y_+=x_0$.

\[prop-ex\] Define $$(\Gamma_+)_c:=\{(y,\omega,E)\in \Gamma_+\ |\ (y-\tau_+(y,\omega)\omega,\omega,E)\in \Gamma_-\}.$$ Then $\Gamma_+\backslash (\Gamma_+)_c$ has zero-measure in $\Gamma$ and there is a continuous extension $\ol{t}$ of $t:D\to {\mathbb{R}}$ onto ${\mathcal{D}}_-:=D\cup \Gamma_-\cup (\Gamma_+)_c$ given by $$\ol{t}(x,\omega)=\begin{cases}
t(x,\omega), & \textrm{if}\ (x,\omega,E)\in D, \\
\tau_+(x,\omega), & \textrm{if}\ (x,\omega,E)\in (\Gamma_+)_c, \\
0, & \textrm{if}\ (x,\omega,E)\in \Gamma_-.
\end{cases}$$

The result is a straightforward consequence of the above Lemmas.

Note that in the case where $G$ is convex and its boundary is $C^1$-regular the above extension $\ol{t}$ is continuous on $\ol G\times S$. For example, for the ball $G=B(0,r)$ we have $\tau_+(x,\omega)=2|{\left\langle}x,\omega{\right\rangle}|$ and $t(x,\omega)={\left\langle}x,\omega{\right\rangle}+\sqrt{{\left\langle}x,\omega{\right\rangle}^2+r^2-{\left\Vert x\right\Vert}^2}$ ([@tervo14 Example 3.1]). We find that for $(y,\omega)\in \Gamma'=\partial G\