We assume that for all $d \geqslant 0$, $\mathbb{P}(X \in d\mathbb{Z}) < 1$, and that $g$ is of class $\mathscr{C}^1$, with support in $[0, K]$ with $K > 0$. The function $f$ is the unique bounded and uniformly continuous solution of equation (E), and $f'$ is bounded and uniformly continuous. Prove that the function $x \mapsto \sup_{t \geqslant x} f'(t)$ admits a finite limit when $x \rightarrow +\infty$. We denote
$$c := \lim_{x \rightarrow +\infty} \sup_{t \geqslant x} f'(t)$$