Lemma: A falling switch region above the ramp

There are $c_0>0$ and a nonempty open $O\subset \{t>\epsilon \}$ on which ${\partial }_{t}g\le -c_0$. Moreover $O\cap P_g=\varnothing$.

Proof. Choose $(t_{\ast },{\mathbf{x}}_{\ast })\in W$, so $t_{\ast }>\epsilon$ and $g(t_{\ast },{\mathbf{x}}_{\ast })=1$. Compact support gives $t_1\in (t_{\ast },T)$ with $g(t_1,{\mathbf{x}}_{\ast })=0$; the mean value theorem gives a point between them with ${\partial }_{t}g<0$, and smoothness gives the open neighbourhood and $c_0$ (XII-C's lemma). On $P_g$ all derivatives of $g$ vanish, so $O\cap P_g=\varnothing$.