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Suppose that f is continuous on (0,1) and that int[0,x] f = int[x,1] f

  1. Aug 2, 2010 #1
    1. The problem statement, all variables and given/known data

    Suppose that f is continuous on (0,1) and that

    int[0,x] f = int[x,1] f

    for all x in [0,1]. Prove that f(x)=0 for all x in [0,1].

    2. Relevant equations

    We know that since f is continuous on (0,1), F(x) = int[0,x] f and F'(x) = f(x) for x in (0,1).

    3. The attempt at a solution

    What I have so far is:

    int[0,x] f = int[x,1] f

    int[0,x] f = int[1,0] f - int[0,x] f

    F(x) = C - F(x) for some constant C

    F'(x) = -F'(x)

    f(x) = - f (x)

    2 f (x) = 0

    f (x)=0 for all x in (0,1).

    But this does not show that f(x)=0 for x=0 and x=1, and I am supposed to show that f(x)=0 for all x in the closed interval [0,1].

    Any hints on how to do this?
     
  2. jcsd
  3. Aug 2, 2010 #2
    Ok, I just found out from my professor that there is a typo in the problem, and that f is continuous on [0,1], so I think I can take it from here.
     
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