Welcome to our community

Be a part of something great, join today!

Real Analysis

ssh

New member
Jun 30, 2012
17
Hi I am post graduate student looking for the answer of the question that appeared in one of the previous papers.

Please help me in solving this.

Thanx in advance

Let α be of bounded variation on [a,b]. let V(n) be the total variation of α on [a,x] and let V(a)=0 . if f Є R(α), prove that f Є R(V)?Let α be of bounded variation on [a,b]. let V(n) be the total variation of α on [a,x] and let V(a)=0 . if f Є R(α), prove that f Є R(V)?
 

Opalg

MHB Oldtimer
Staff member
Feb 7, 2012
2,725
Hi ssh, and welcome to Math Help Boards.

Let α be of bounded variation on [a,b]. let V(n) be the total variation of α on [a,x] and let V(a)=0 . if f Є R(α), prove that f Є R(V)?
There seems to be some unexplained notation here. First, I assume that V(n) should be V(x)? Second, what does the notation R(a) and R(V) mean?
 

ssh

New member
Jun 30, 2012
17
Yes Vn(x) is Total Variation of x
F Є R(α) and f Є R(V) means f is reimann integrable on α and V respectively
 

ssh

New member
Jun 30, 2012
17
Is my question unclear or unanswerable?
 

Opalg

MHB Oldtimer
Staff member
Feb 7, 2012
2,725
Is my question unclear or unanswerable?
The question is still totally unclear to me. You start by saying "Let α be of bounded variation on [a,b]", from which I understand that α must be a function (in fact, a function of bounded variation). But then you say "F Є R(α) ... means f is reimann integrable on α", which makes no sense at all. How can a function be Riemann integrable on a function?

You also say "f Є R(V) means f is reimann integrable on ... V". That can only make sense if V is a set. But no set V has been defined.

Here is my version of the question that I think you may be trying to ask:
Let f be a function of bounded variation on the interval [a,b]. For each x in [a,b], let V(f)(x) be the total variation of f on the interval [a,x]. Prove that if f is Riemann integrable on [a,b] then so is the function V(f).

If that is the correct version of the question, let us know and perhaps someone here will be able to help.
 

ssh

New member
Jun 30, 2012
17
This is the question i saw in one of the question papers for which i couldnt find an answer.

Let α be of bounded variation on [a,b]. let V(n) be the total variation of α on [a,x] and let V(a)=0 . if f Є R(α), prove that f Є R(V)?

If f Є R(α ) then U(P,f,α) – L(P,f,α) < ε right which is the sufficient condition for f to be Reimann integrable on α. (Atleast my book says so, correct me if this is wrong). Now if we want to prove f Є R (V), we should be showing that U(P, f, V) – L(P,f, V) < Є1 (say), but how? Here V is the total variation of x on α in the interval [a,x] also given that V(a) = 0 how does this help. The Proof for this in my book is not clear.
 

ssh

New member
Jun 30, 2012
17
My question remains unanswered wherever i tried?