I'm finding some trouble in showing this fact... It seems to me that I should go trough riemann approximations of the integral, but i cannot formalize this process. That's why i'm asking for some help, at least for some hints...
Let $f:[0,1]\to\mathbb R$ be a function with bounded variation, and let $TV_0^1(f)$ be its total variation. Then show that for any $h\in (0,1)$, the following holds: $\int_0^{1-h}|f(x+h)-f(x)|\mathrm dx\leq hTV_0^1(f).$ Thanks for your attention