Def.The variation of $f$ on $[a, b]$ corresponding to a partition $P = \{x_0, x_1, \ldots, x_n\},$$V^P(f):=\sum_{i=1}^{n} |f(x_i) - f(x_{i-1})|.$Obviously, if $Q$ is a refinement of $P$, then $V^P(f) \leq V^Q(f)$. If the set $\{V^P(f) : P \text{ is a partition of } [a, b]\}$ is bounded, then $f$ is called a function of bounded variation on $[a, b]$ and $V^{[a, b]}(f) := \sup_P V^P(f)$ is the total variation of $f$ on $[a, b]$.
Suppose that $f$ is of bounded variation on $(-\infty, \infty)$ and$\int_{-\infty}^{\infty} |f(x)| \,dx < \infty,$do we have $\lim_{x \to \pm \infty} f(x) = 0 ?$