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: Proving a f(p) = 0 for all p.

  1. Nov 19, 2009 #1
    1. The problem statement, all variables and given/known data

    If D is open, and if f is continuous, bounded, and obeys f(p)>or=0 for all p in D, then the double integral over D of f is equal to 0 implies f(p) = 0 for all p.

    2. Relevant equations

    Hint: There is a neighborhood where f(p)>or=d.

    3. The attempt at a solution

    The integral is equal to the sum of f(p)*A(Rij) for some Rij partition of f(x). Since Rij > 0, for the whole thing to be 0, f(p) has to be 0.

    Not really sure how to start my proof.

    f is continuous and m<or= f(p) <or= M so mA(D) <or= double integral <or= MA(D)

    since the double integral is = 0,

    mA(D) <or= 0 <or= MA(D) which means m=M=0 and f(p) = 0 for all p.

    Is this correct?
     
  2. jcsd
  3. Nov 19, 2009 #2
    To my eye, an indirect proof looks tempting. D is open so it's measure is not zero. Since f is continuous, for it to be nonzero at a point p means also that there exists a neighbourhood where f is nonzero which is a subset of D (since D is open), and since the neighbourhoods are open sets, they have nonzero measure and therefore the integral can't be zero.
     
Share this great discussion with others via Reddit, Google+, Twitter, or Facebook