> >Hello All,

> >The well-known definition of bounded variation functions is about
> >their behavior on closed intervals.
> >To say, "The total variation of real-valued function f, defined on an
> >interval [a, b] belongs to R is the quantity V_a_b_(f) = sup(sum(|f(x_i
> >+1) - f(x_i)|)) of all partitions of the interval considered", (etc.)

> >My question is, can I use or define total variation for a half-open
> >interval, say, [a, b)?

> Yes.

> >In any case, my intention is to define (or use) the following: for any
> >eps > 0 the total variation of the function
> >f on the interval [a, b - eps] is finite. Is it the same as to say
> >that the total variation of a function over the half-open interval [a,
> >b) is finite?

> Of course not - the total variation on [a, b-eps] could be finite
> for every eps > 0 but tend to infinity as eps -> 0.

> If the total variation on [a,b-eps] is _bounded_ for eps > 0
> then the total variation on [a,b) is finite.

> >Regards,
> >Miki

> David C. Ullrich

> "Understanding Godel isn't about following his formal proof.
> That would make a mockery of everything Godel was up to."
> (John Jones, "My talk about Godel to the post-grads."
> in sci.logic.)

> Yes.