# Analysis (riemann integral) question

1. Jun 27, 2009

### fraggle

denote (x knot as x_0)

Suppose α(x) increases on [a,b] a≤ x_0 ≤b, α is continuous at x_0,
f(x_0) =1 , at all other x in [a,b] f(x)=0.

Prove that f is Riemann Integrable and that ∫fdα=0.

Can anyone suggest a good method to show that inf U(P,f,α)= 0?
I have proved that the lower limit is equal to zero. Now I just need to prove that the upper limit is equal to zero or that f is Riemann Integrable.

Here's my attempt for the former:

α continuous at x_0 ⇒ for each ε>0 there exists a δ>0 s.t for q in [a,b] if
⎮α(x_0)-α(q)⎮<ε then ⎮x_0 - q⎮<δ
Pick elements p<x_0<q in the neighborhood of radius δ about x_0 we can then choose a partition such that
Δα_i=(α(q)-α(p))/n
this is true for any segment (x_i-1,x_i) s.t ⎮x_i -x_0⎮<δ
Now choose a partition P of [a,b] with the above partition in the neighborhood of x_0 and arbitrarily let a=x_1 and b=b_2.
The definition of f(x) implies that the only segment of the partition P where Σsupf(x) is not equal to zero is a segment in the neighborhood of radius δ about x_0.
There supf(x)=1
so Σsupf(x)Δα_i = (α(q) -α(p))/n
This being true for all n in N we can take n very large to get zero.

Does this work?
If not can anyone give a hint?
Thank you

Last edited: Jun 27, 2009