Let $u: \mathbb{R} \to \mathbb{R}$ such that $u \in C^1 (\mathbb{R}) \cap \mathrm{BV} (\mathbb{R})$. Prove that
$$\frac{1}{\varepsilon} \int_{-\infty}^{\infty} |u(x+\varepsilon) - u (x)| dx \leq TV (u), $$
where $TV(u)$ is the total variation of u.
Any idea?