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)?
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?
> 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)?
> 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?
> Regards, > Miki
Try with f(x) = 1/x and (0,1]: for any e > 0 the total variation of f (x) in [e,1] is finite but not so in (0,1]
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.)
> 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.)
Thanks,
My question is, can I use or define total variation for a half-open
> >interval, say, [a, b)?
> Yes.
So, what is the definition of total variation on the interval [a, b)? How can I write the sum if the left-end of the interval is open?
If I take the function f(x)=1/(x-5) for example, it is of bounded variation on [4, 5 - e] for every e > 0 as long as e does not tends to zero (so actually, its not for every e...) OK, so what about saying that it is of bounded variation on [4, 5). Now is it correct? If so, whats the meaning of it in terms of total variation? how can I compute its total variation?
>On Nov 7, 4:18 pm, David C. Ullrich <dullr...@sprynet.com> wrote: >> 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.)
>Thanks,
>My question is, can I use or define total variation for a half-open >> >interval, say, [a, b)?
>> Yes.
>So, what is the definition of total variation on the interval [a, b)? >How can I write the sum if the left-end of the interval is open?
If I is any interval (including [a,b], (a,b), R, etc) and f is defined on I then the total variation of f on I is the sup of
V(f,I) = sum |f(t_j) - f(t_{j-1}|
over all choices of numbers t_0, .., t_n in I with t_{j-1} < t_j.
And then we say that f has bounded variation on I if V(f,I) is finite.
>If I take the function f(x)=1/(x-5) for example, >it is of bounded variation on [4, 5 - e] for every e > 0 as long as e >does not tends to zero (so actually, its not for every e...)
Yes, for every e. You're confused about the difference between "finite" and "bounded".
For _every_ e > 0 (well, we also want e < 1) we have
V(f, [4, 5-e]) < infinity.
But V(f, [4, 5-e]) is not bounded, and in particular
V(f, [4,5)) = infinity.
>OK, so what about saying that it is of bounded variation on [4, 5). >Now is it correct?
No.
> If so, whats the meaning of it in terms of >total variation? how can I compute its total variation?
From the definition above. The computation is very simple here because f is monotone.
> On Sat, 7 Nov 2009 11:01:25 -0800 (PST), miki <miki.li...@gmail.com> > wrote:
> >On Nov 7, 4:18 pm, David C. Ullrich <dullr...@sprynet.com> wrote: > >> 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.)
> >Thanks,
> >My question is, can I use or define total variation for a half-open > >> >interval, say, [a, b)?
> >> Yes.
> >So, what is the definition of total variation on the interval [a, b)? > >How can I write the sum if the left-end of the interval is open?
> If I is any interval (including [a,b], (a,b), R, etc) and f is defined > on I then the total variation of f on I is the sup of
> V(f,I) = sum |f(t_j) - f(t_{j-1}|
> over all choices of numbers t_0, .., t_n in I with t_{j-1} < t_j.
> And then we say that f has bounded variation on I if V(f,I) is finite.
> >If I take the function f(x)=1/(x-5) for example, > >it is of bounded variation on [4, 5 - e] for every e > 0 as long as e > >does not tends to zero (so actually, its not for every e...)
> Yes, for every e. You're confused about the difference between > "finite" and "bounded".
> For _every_ e > 0 (well, we also want e < 1) we have
> V(f, [4, 5-e]) < infinity.
> But V(f, [4, 5-e]) is not bounded, and in particular
> V(f, [4,5)) = infinity.
> >OK, so what about saying that it is of bounded variation on [4, 5). > >Now is it correct?
> No.
> > If so, whats the meaning of it in terms of > >total variation? how can I compute its total variation?
> From the definition above. The computation is very simple here > because f is monotone.
> >thanks > >Miki- Hide quoted text -
> - Show quoted text -- Hide quoted text -
> - Show quoted text -
Thanks. I did confuse between "finite" and "bounded". Well, I care only for finite. Nevertheless, I think I have a confusion also about the open and closed intervals. Meaning,
"for every e > 0 ... "
You wrote:
1. V(f, [4, 5-e]) < infinity. I ask why. Namely, If I say for "every e > 0" it means that I can take e to be as small as I want, so I can take also the limit toward zero. isn't it? Of course, in the limit we have problems so how can the total variation be not infinite.
2. V(f, [4, 5)) = infinity. This case is understood.
2. V(f, [4, 5-e)) = infinity. What about this expression? Is this as no. 2? What is the difference from no. 1?
what about this case: f(x) = x Is th total variation of f(x) over [0, 1) is finite now? why?
>On Nov 7, 11:42 pm, David C. Ullrich <ullr...@math.okstate.edu> wrote: >> On Sat, 7 Nov 2009 11:01:25 -0800 (PST), miki <miki.li...@gmail.com> >> wrote:
>> >On Nov 7, 4:18 pm, David C. Ullrich <dullr...@sprynet.com> wrote: >> >> 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.)
>> >Thanks,
>> >My question is, can I use or define total variation for a half-open >> >> >interval, say, [a, b)?
>> >> Yes.
>> >So, what is the definition of total variation on the interval [a, b)? >> >How can I write the sum if the left-end of the interval is open?
>> If I is any interval (including [a,b], (a,b), R, etc) and f is defined >> on I then the total variation of f on I is the sup of
>> V(f,I) = sum |f(t_j) - f(t_{j-1}|
>> over all choices of numbers t_0, .., t_n in I with t_{j-1} < t_j.
>> And then we say that f has bounded variation on I if V(f,I) is finite.
>> >If I take the function f(x)=1/(x-5) for example, >> >it is of bounded variation on [4, 5 - e] for every e > 0 as long as e >> >does not tends to zero (so actually, its not for every e...)
>> Yes, for every e. You're confused about the difference between >> "finite" and "bounded".
>> For _every_ e > 0 (well, we also want e < 1) we have
>> V(f, [4, 5-e]) < infinity.
>> But V(f, [4, 5-e]) is not bounded, and in particular
>> V(f, [4,5)) = infinity.
>> >OK, so what about saying that it is of bounded variation on [4, 5). >> >Now is it correct?
>> No.
>> > If so, whats the meaning of it in terms of >> >total variation? how can I compute its total variation?
>> From the definition above. The computation is very simple here >> because f is monotone.
>> >thanks >> >Miki- Hide quoted text -
>> - Show quoted text -- Hide quoted text -
>> - Show quoted text -
>Thanks. >I did confuse between "finite" and "bounded". Well, I care only for >finite. >Nevertheless, I think I have a confusion also about the open and >closed intervals.
No, looking below it seems the confusion is more basic than that.
>Meaning,
>"for every e > 0 ... "
>You wrote:
>1. V(f, [4, 5-e]) < infinity. >I ask why. Namely, If I say for "every e > 0" it means that I can take >e to be as small as I want, so I can take also the limit toward zero. >isn't it?
No, it doesn't mean that!
If the V(f, [4, 5-e]) were _bounded_ for e > 0 then you could say that. But just supposing that V(f, [4, 5-e]) is finite for every e > 0 says nothing about boundedness and also nothing about the limit.
Look. Is the following true or false?
(i) e > 0 for every e > 0.
I hope you agree that's true - if (i) is false then you have to give me an example of an e > 0 such that e is not > 0.
Now according to you, since (i) is true for every e > 0 it follows that we can take the limit. We take the limit as e -> 0 and we get the following statement:
(ii) 0 > 0.
Which is false.
That example has nothing to do with bounded variation. The point to the example is to convince you that what you're confused about _also_ has nothing to do with bounded variation - the things you think about how "of course if this is true then we can take the limit" are simply not so.
>Of course, in the limit we have problems so how can the >total variation be not infinite.
>2. V(f, [4, 5)) = infinity. >This case is understood.
>2. V(f, [4, 5-e)) = infinity. >What about this expression? Is this as no. 2? What is the difference >from no. 1?
>what about this case: f(x) = x >Is th total variation of f(x) over [0, 1) is finite now? why?
>Thanks again, >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.)
> On Sat, 7 Nov 2009 22:20:38 -0800 (PST), miki <miki.li...@gmail.com> > wrote:
> >On Nov 7, 11:42 pm, David C. Ullrich <ullr...@math.okstate.edu> wrote: > >> On Sat, 7 Nov 2009 11:01:25 -0800 (PST), miki <miki.li...@gmail.com> > >> wrote:
> >> >On Nov 7, 4:18 pm, David C. Ullrich <dullr...@sprynet.com> wrote: > >> >> 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.)
> >> >Thanks,
> >> >My question is, can I use or define total variation for a half-open > >> >> >interval, say, [a, b)?
> >> >> Yes.
> >> >So, what is the definition of total variation on the interval [a, b)? > >> >How can I write the sum if the left-end of the interval is open?
> >> If I is any interval (including [a,b], (a,b), R, etc) and f is defined > >> on I then the total variation of f on I is the sup of
> >> V(f,I) = sum |f(t_j) - f(t_{j-1}|
> >> over all choices of numbers t_0, .., t_n in I with t_{j-1} < t_j.
> >> And then we say that f has bounded variation on I if V(f,I) is finite.
> >> >If I take the function f(x)=1/(x-5) for example, > >> >it is of bounded variation on [4, 5 - e] for every e > 0 as long as e > >> >does not tends to zero (so actually, its not for every e...)
> >> Yes, for every e. You're confused about the difference between > >> "finite" and "bounded".
> >> For _every_ e > 0 (well, we also want e < 1) we have
> >> V(f, [4, 5-e]) < infinity.
> >> But V(f, [4, 5-e]) is not bounded, and in particular
> >> V(f, [4,5)) = infinity.
> >> >OK, so what about saying that it is of bounded variation on [4, 5). > >> >Now is it correct?
> >> No.
> >> > If so, whats the meaning of it in terms of > >> >total variation? how can I compute its total variation?
> >> From the definition above. The computation is very simple here > >> because f is monotone.
> >> >thanks > >> >Miki- Hide quoted text -
> >> - Show quoted text -- Hide quoted text -
> >> - Show quoted text -
> >Thanks. > >I did confuse between "finite" and "bounded". Well, I care only for > >finite. > >Nevertheless, I think I have a confusion also about the open and > >closed intervals.
> No, looking below it seems the confusion is more basic than that.
> >Meaning,
> >"for every e > 0 ... "
> >You wrote:
> >1. V(f, [4, 5-e]) < infinity. > >I ask why. Namely, If I say for "every e > 0" it means that I can take > >e to be as small as I want, so I can take also the limit toward zero. > >isn't it?
> No, it doesn't mean that!
> If the V(f, [4, 5-e]) were _bounded_ for e > 0 then you > could say that. But just supposing that V(f, [4, 5-e]) is > finite for every e > 0 says nothing about boundedness > and also nothing about the limit.
> Look. Is the following true or false?
> (i) e > 0 for every e > 0.
> I hope you agree that's true - if (i) is false then > you have to give me an example of an e > 0 such > that e is not > 0.
> Now according to you, since (i) is true for every e > 0 > it follows that we can take the limit. We take the limit > as e -> 0 and we get the following statement:
> (ii) 0 > 0.
> Which is false.
> That example has nothing to do with bounded variation. > The point to the example is to convince you that what > you're confused about _also_ has nothing to do with > bounded variation - the things you think about how "of > course if this is true then we can take the limit" are > simply not so.
> >Of course, in the limit we have problems so how can the > >total variation be not infinite.
> >2. V(f, [4, 5)) = infinity. > >This case is understood.
> >2. V(f, [4, 5-e)) = infinity. > >What about this expression? Is this as no. 2? What is the difference > >from no. 1?
> >what about this case: f(x) = x > >Is th total variation of f(x) over [0, 1) is finite now? why?
> >Thanks again, > >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.)- Hide quoted text -
> - Show quoted text -- Hide quoted text -
> - Show quoted text -
I agree with all your arguments. It is difficult to understand such kind of content via the internet, so I tend to ask some questions like no 1. just to be sure.
So, what about the case of no. 2. where we talk about the interval [4, 5-e) and about the no. 3 where the interval is [4, 5). I remind you that the function is f(x) = 1/(x-5)
and I am interesting only with the total variation being finite for every epsilon on the mentioned intervals. I dont care for boundness at all.
I guess that the case for half open interval is different ... is it? if so why? I guess that there is a different between [4, 5-e) (for e > 0) and the case of [4, 5).
You know what, I think I know the answer, please just confirm it.
To say, The case of [4, 5-e) is like case of [4, 5-e] (e > 0), namely, the total variation is finite for every e > 0.
The case of [4, 5) is problematic since by the definition of total variation: it is the sup of all partitions, so its kind of taking the limit on the interval [4, 5) on all the partitions, so I can build a strictly monotone increasing set of sum over all the partitions and take limit which is infinity, so ... the total variation is infinite. Is it correct?
>On Nov 8, 3:54 pm, David C. Ullrich <dullr...@sprynet.com> wrote: >> On Sat, 7 Nov 2009 22:20:38 -0800 (PST), miki <miki.li...@gmail.com> >> wrote:
>> >On Nov 7, 11:42 pm, David C. Ullrich <ullr...@math.okstate.edu> wrote: >> >> On Sat, 7 Nov 2009 11:01:25 -0800 (PST), miki <miki.li...@gmail.com> >> >> wrote:
>> >> >On Nov 7, 4:18 pm, David C. Ullrich <dullr...@sprynet.com> wrote: >> >> >> 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.)
>> >> >Thanks,
>> >> >My question is, can I use or define total variation for a half-open >> >> >> >interval, say, [a, b)?
>> >> >> Yes.
>> >> >So, what is the definition of total variation on the interval [a, b)? >> >> >How can I write the sum if the left-end of the interval is open?
>> >> If I is any interval (including [a,b], (a,b), R, etc) and f is defined >> >> on I then the total variation of f on I is the sup of
>> >> V(f,I) = sum |f(t_j) - f(t_{j-1}|
>> >> over all choices of numbers t_0, .., t_n in I with t_{j-1} < t_j.
>> >> And then we say that f has bounded variation on I if V(f,I) is finite.
>> >> >If I take the function f(x)=1/(x-5) for example, >> >> >it is of bounded variation on [4, 5 - e] for every e > 0 as long as e >> >> >does not tends to zero (so actually, its not for every e...)
>> >> Yes, for every e. You're confused about the difference between >> >> "finite" and "bounded".
>> >> For _every_ e > 0 (well, we also want e < 1) we have
>> >> V(f, [4, 5-e]) < infinity.
>> >> But V(f, [4, 5-e]) is not bounded, and in particular
>> >> V(f, [4,5)) = infinity.
>> >> >OK, so what about saying that it is of bounded variation on [4, 5). >> >> >Now is it correct?
>> >> No.
>> >> > If so, whats the meaning of it in terms of >> >> >total variation? how can I compute its total variation?
>> >> From the definition above. The computation is very simple here >> >> because f is monotone.
>> >> >thanks >> >> >Miki- Hide quoted text -
>> >> - Show quoted text -- Hide quoted text -
>> >> - Show quoted text -
>> >Thanks. >> >I did confuse between "finite" and "bounded". Well, I care only for >> >finite. >> >Nevertheless, I think I have a confusion also about the open and >> >closed intervals.
>> No, looking below it seems the confusion is more basic than that.
>> >Meaning,
>> >"for every e > 0 ... "
>> >You wrote:
>> >1. V(f, [4, 5-e]) < infinity. >> >I ask why. Namely, If I say for "every e > 0" it means that I can take >> >e to be as small as I want, so I can take also the limit toward zero. >> >isn't it?
>> No, it doesn't mean that!
>> If the V(f, [4, 5-e]) were _bounded_ for e > 0 then you >> could say that. But just supposing that V(f, [4, 5-e]) is >> finite for every e > 0 says nothing about boundedness >> and also nothing about the limit.
>> Look. Is the following true or false?
>> (i) e > 0 for every e > 0.
>> I hope you agree that's true - if (i) is false then >> you have to give me an example of an e > 0 such >> that e is not > 0.
>> Now according to you, since (i) is true for every e > 0 >> it follows that we can take the limit. We take the limit >> as e -> 0 and we get the following statement:
>> (ii) 0 > 0.
>> Which is false.
>> That example has nothing to do with bounded variation. >> The point to the example is to convince you that what >> you're confused about _also_ has nothing to do with >> bounded variation - the things you think about how "of >> course if this is true then we can take the limit" are >> simply not so.
>> >Of course, in the limit we have problems so how can the >> >total variation be not infinite.
>> >2. V(f, [4, 5)) = infinity. >> >This case is understood.
>> >2. V(f, [4, 5-e)) = infinity. >> >What about this expression? Is this as no. 2? What is the difference >> >from no. 1?
>> >what about this case: f(x) = x >> >Is th total variation of f(x) over [0, 1) is finite now? why?
>> >Thanks again, >> >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.)- Hide quoted text -
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>I agree with all your arguments. >It is difficult to understand such kind of content via the internet, >so I tend to ask some questions like no 1. just to be sure.
>So, what about the case of no. 2. where we talk about the interval [4, >5-e) >and about the no. 3 where the interval is [4, 5). >I remind you that the function is f(x) = 1/(x-5)
What about this case? I've already answered all the questions you had about this.
>and I am interesting only with the total variation being finite for >every epsilon on the mentioned >intervals. I dont care for boundness at all.
>I guess that the case for half open interval is different ... is it? >if so why? >I guess that there is a different between [4, 5-e) (for e > 0) and the >case of [4, 5).
>You know what, I think I know the answer, please just confirm it.
>To say, The case of [4, 5-e) is like case of [4, 5-e] (e > 0), namely, >the total variation is finite >for every e > 0.
>The case of [4, 5) is problematic since by the definition of total >variation: it is the sup >of all partitions, so its kind of taking the limit on the interval [4, >5) on all the partitions, so I can build a strictly monotone >increasing set of sum over all the partitions and take limit which is >infinity, so ... the total variation is infinite. Is it correct?
>Thanks a lot, >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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