In this subsection, we still assume that $J_n = J_n^{(\mathrm{C})}$. Moreover, we assume that $\beta = 1$ and $h = 0$. We are given $x \in \mathbb{R}$ and $\varepsilon > 0$. Let $k$ be a non-zero natural integer such that $k \geqslant \frac{2}{\varepsilon Z_\infty}$. We define the function $$f_k : u \in \mathbb{R} \longmapsto \begin{cases} 1 & \text{if } u \leqslant x \\ 1 - k(u-x) & \text{if } x < u \leqslant x + \frac{1}{k} \\ 0 & \text{otherwise} \end{cases}$$ Show that $f_k$ is $k$-Lipschitz on $\mathbb{R}$.
In this subsection, we still assume that $J_n = J_n^{(\mathrm{C})}$. Moreover, we assume that $\beta = 1$ and $h = 0$.
We are given $x \in \mathbb{R}$ and $\varepsilon > 0$. Let $k$ be a non-zero natural integer such that $k \geqslant \frac{2}{\varepsilon Z_\infty}$. We define the function
$$f_k : u \in \mathbb{R} \longmapsto \begin{cases} 1 & \text{if } u \leqslant x \\ 1 - k(u-x) & \text{if } x < u \leqslant x + \frac{1}{k} \\ 0 & \text{otherwise} \end{cases}$$
Show that $f_k$ is $k$-Lipschitz on $\mathbb{R}$.