This is a follow-up question to Darboux coordinates for contact geometry since I couldn't comment on the posts of other people yet. In the first the answer of C. Falcon (thanks a lot for your detailed answer! It clarified a lot of my questions.) I don't understand why $t\mapsto\varepsilon_t$ is a lower semicontinuous function, where C. Falcon defined:
$\qquad \varepsilon_t\doteq \mathrm{sup}\{\varepsilon>0:B(0,\varepsilon)\subseteq U_{t}\}$
In the context of the question $U_t$ is the domain of the smooth funtion $H_t:U_t\rightarrow\mathbb{R}$ that solves $\dot\alpha_t(R_{\alpha_t})+dH_t(R_{\alpha_t})=0$ with respect to a given smooth family of contact forms $\alpha_t$, $t\in[0,1]$ and $R_{\alpha_t}$ their corresponding Reeb Vectorfields. (As far as I've understood; the open subset $U_t\subseteq\mathbb{R}$ is the domain of the chart in which $R_{\alpha_t}$ is equal to $\frac{\partial}{\partial x_1}$.)
Thank you!
