grandes-ecoles 2025 Q34

grandes-ecoles · France · centrale-maths2__official Proof Proof That a Map Has a Specific Property
In this subsection, we assume that $J_n = J_n^{(\mathrm{C})}$, the matrix introduced in subsection A-II.
We set $G_h : x \longmapsto \frac{(x-h)^2}{2\beta} - \ln(2\operatorname{ch}(x))$. We now assume that $h > 0$.
We set $\gamma_h = G_h''(u_h)$ and we denote $f_h : x \longmapsto \frac{\widehat{G}_h(x)}{x^2}$. Show that $f_h$ can be extended to a continuous function on all of $\mathbb{R}$ by setting $f_h(0) = \frac{\gamma_h}{2}$. 
In this subsection, we assume that $J_n = J_n^{(\mathrm{C})}$, the matrix introduced in subsection A-II.

We set $G_h : x \longmapsto \frac{(x-h)^2}{2\beta} - \ln(2\operatorname{ch}(x))$. We now assume that $h > 0$.

We set $\gamma_h = G_h''(u_h)$ and we denote $f_h : x \longmapsto \frac{\widehat{G}_h(x)}{x^2}$. Show that $f_h$ can be extended to a continuous function on all of $\mathbb{R}$ by setting $f_h(0) = \frac{\gamma_h}{2}$.
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