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Homework Help: A analysis problem

  1. Feb 20, 2010 #1
    1. The problem statement, all variables and given/known data
    [tex]f[/tex] is continuous on [a,b]
    [tex]f_{1}(x)=\int^x_a f(t)dt[/tex]
    [tex]f_{2}(x)=\int^x_a f_{1}(t)dt [/tex]
    .....
    [tex]\forall x\in[a,b],\exists n[/tex] depends on x , such that [tex]f_{n}(x)=0.[/tex]
    prove that [tex]f\equiv0.[/tex]

    2. Relevant equations
    mathematical analysis
    3. The attempt at a solution
    copy the taylor theorem 's proof???
    but I get nothing.
     
  2. jcsd
  3. Feb 21, 2010 #2

    HallsofIvy

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    Science Advisor

    If, for some n, fn is identically 0, then it is a constant and so its derivative is identically equal to 0. But, by the fundamental theorem, the derivative of [itex]f_n(x)= \int_a^x f_{n-1}(t)dt[/itex] is fn-1(x). Therefore, if fn(x) is identically 0 on [a, b], so is fn-1(x).
     
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