1. Limited time only! Sign up for a free 30min personal tutor trial with Chegg Tutors
    Dismiss Notice
Dismiss Notice
Join Physics Forums Today!
The friendliest, high quality science and math community on the planet! Everyone who loves science is here!

Homework Help: Riemmann/Darboux-Integral Question

  1. Apr 4, 2008 #1
    1. The problem statement, all variables and given/known data

    (i) Suppose that F is continuous on [a,b] and [tex]\int_a^b FG = 0[/tex] for all continuous functions G on [a,b]. Prove that F = 0.

    (ii) Suppose now that G(a) = G(b) = 0. Does it again follow that F must be identically zero?

    2. Relevant equations
    Let P be any partition of [a,b]

    Upper Darboux
    [tex]U(F,P) = \sum_{k=0}^\n sSupF(x_k)(t_k - t_{k-1})[/tex]
    Lower Darboux
    [tex]L(F,P) = \sum_{k=0}^\n iInfF(x_k)(t_k - t_{k-1})[/tex]

    F is integrable iff
    inf U(F,P) = U(F) = L(F) = sup L(F,P)

    3. The attempt at a solution

    For (i) I argued the following.

    Let G = F.
    Clear H = F*F >= 0 and is continuous.
    Let's say H(y) > 0.
    So I can find an interval (y - c, y + c) because H is continuous, where H(x) > H(y)/2.
    Clearly U(H,P) > H(y)/2*c
    Which means U(H) >= H(y)/2*c > 0.

    Thus F must be identically zero.

    For (ii), I said it still needed to be identically zero, and I'm not certain how the argument needs to be changed. Perhaps if F was only nonzero very near a and b, but because of the continuity of F, wouldn't that still bee a problem?
  2. jcsd
  3. Apr 4, 2008 #2


    User Avatar
    Science Advisor
    Homework Helper

    How about G(x)=F(x)*(x-a)*(b-x)? There's nothing special about (x-a)*(b-x), it's just a positive continuous function on [a,b] that vanishes only at the endpoints.
  4. Apr 4, 2008 #3
    Thanks Dick, works perfectly. =)
Share this great discussion with others via Reddit, Google+, Twitter, or Facebook