Definition of the total variation of a function $g:\mathbb{R}\to\mathbb{R}$

Question

if the total variation of a a real function $f:[a,b]\to \mathbb{R}$ over $\textbf{P}=\{a=t_0<t_1<...<t_m=b\}$ is

$$ V^{a}_{b}(f)=\sup_{\textbf{P}}V(f,\textbf{P}) $$ where $$ V(f,\textbf{P})=\sum^{m}_{i=1}|f(t_i)-f(t_{i-1})| $$ then what is the total variation of a function $g:\mathbb{R}\to\mathbb{R}$? I didn't find a definition of this case.

Answer 1

Do the same thing you do with integrals: Take the (independent) limits as $a \rightarrow -\infty$ and $b \rightarrow \infty$.