# Real Analysis

#### ssh

##### New member
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
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
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
Is my question unclear or unanswerable?

#### Opalg

##### MHB Oldtimer
Staff member
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
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
My question remains unanswered wherever i tried?