Let $f:\mathbb{R} \to \mathbb{R}$ be a strictly increasing bounded continuous function and let $\{ t_n \}$ be a convergent to $0$ sequence of real numbers. Put $f_n(x):=f(x+t_n)$, $n=1,2,...$
Is then $\mathrm{TV}(f_n - f) \to 0$ as $n \to \infty$, where $\mathrm{TV}$ is the total variation?