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How can one define the total variation of a function of bounded variation on $\Bbb R$?

i.e., how one can evaluate the total variation on infinite intervals?!

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    See http://math.stackexchange.com/questions/116196/functions-of-bounded-variation-on-all-mathbbr/116320#116320 for some properties of these functions.2012-10-18

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Well, I would do something like this. If say $f(x) \in BV[a,b]\quad\forall [a,b]\subset\mathbf{R}$ then you could define its TV as something like $\lim\limits_{r\rightarrow+\infty}V_{-r}^r(f)$ (if it exists), somewhat close to v.p. of an integral, but that's kind of an ad-hoc suggestion.

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    Fine, good idea Thanks2012-10-21