Question on bounded variation functions - sci.math

Message from discussion Question on bounded variation functions

David C. Ullrich

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More options Nov 7 2009, 2:18 pm

Newsgroups: sci.math

From: David C. Ullrich <dullr...@sprynet.com>

Date: Sat, 07 Nov 2009 08:18:22 -0600

Local: Sat, Nov 7 2009 2:18 pm

Subject: Re: Question on bounded variation functions

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On Sat, 7 Nov 2009 01:18:24 -0800 (PST), miki <miki.li...@gmail.com>
wrote:

>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.)

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