In this subsection, we assume that $J_n = J_n^{(\mathrm{C})}$, the matrix introduced in subsection A-II.
We set $G_h : x \longmapsto \frac{(x-h)^2}{2\beta} - \ln(2\operatorname{ch}(x))$.
Show that $G_0'$ establishes a continuous bijection from $[u_0; +\infty[$ to $\mathbb{R}_+$. Deduce that the function $u : h \longmapsto u_h$ is continuous on $\mathbb{R}_+$ and differentiable on $\mathbb{R}_+^*$.