Proving That f Є R(V) When f Є R(α)

[ssh]

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]

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]

Yes V_n(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]

Is my question unclear or unanswerable?

 

[Opalg]

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

 

[ssh]

This is the question i saw in one of the question papers for which i couldn't 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]

My question remains unanswered wherever i tried?