Let $f=g-h$ with $g,h\in{\overline{\mathscr{P}}}(N_{\mathbb{R}})$. The effective domain of the recessions of $g$ and of $h$ is the whole of $N_{\mathbb{R}}$. By [Roc70, Theorem 10.5], both $g$ and $h$ are Lipchitzians, hence so is $f$. ∎