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Analysis (riemann integral) question

  1. Jun 27, 2009 #1
    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
  2. jcsd
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