Consider $F:\mathbb{R}\rightarrow\mathbb{R}$ such that $\sup_{a,b}T_F (a,b)<\infty$ where $T_F (a,b)$ is the total variation of $F$ on the interval $[a,b]$. Then we have
i) $\int_\mathbb{R}|F(x+h)-F(x)|dx\leq A |h| $, for some constant $A$ and $\forall h\in \mathbb{R}$;
ii)$\int_\mathbb{R}F(x)\phi'(x)dx\leq A $ whenever $\phi \in C^1$ with compact support and $|\phi|_\infty\leq 1$.