I know what it means for a function $f$ on a bounded interval $[a,b]$ to be of bounded variation. But what does it mean for $f$ to be of BV on $\mathbb{R}$? Does it mean that $\lim_{N\rightarrow\infty} V[-N,N] < \infty$? Here $V$ is the total variation of $f$.
On an interval $[a,b]$, $f$ is of bounded variation if $\sup_\Gamma \sum_{i=1}^m |f(x_i) - f(x_{i-1})| < \infty$ where the supremum is over partitions $\Gamma=\{x_0 < x_1<...<x_m \}$ of $[a,b]$.
It means that $$\sup\sum_{i=1}^n|f(x_i)-f(x_{i-1})|<\infty,$$ where the supremum is taken over all possible $x_0<\dots<x_n$. And yes it is equivalent to the limit condition you wrote.