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Integral help needed

  1. Jul 9, 2008 #1
    1. The problem statement, all variables and given/known data
    PRove that, if f is continuous on [a,b] and the integral from a to b of |f(x)|dx = 0 (thats absolute value of f(x) ), then f(x) = 0 for all x in [a,b]

    2. Relevant equations

    3. The attempt at a solution

    HEre is my attempt....its not very rigorous which is why I'm asking for any tips that could make this a more formal proof:

    Proof by contradiction:
    Assume f(x) is not equal to zero for all x in [a,b]
    then there exists an x1 element of [a,b] such that f(x1) > 0
    Then by properties of integrals we can integrate f(x) from a to x1 and from x1 to b and add them together
    since f(x1) is greater than zero and not just a removable discontinuity (since f is stated as continusous), the area bound by x = a, x = x1, y = 0 and y = f(x) must also be greater than zero and therefore the integral from a to x1 of |f(x)|dx is also greater than zero. Since the function |f(x)| is always nonnegative, the integral from x1 to b of |f(x)|dx is also nonnegative. Since the integral from a to x1 is positive and from x1 to b is non-negative, then their sum must also be positive. But this congradicts the assumption that the integral from a to b of |f(x)|dx = 0 for all x in [a,b] , so f(x) must be 0 for all x in [a,b]
  2. jcsd
  3. Jul 10, 2008 #2


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    Your first statement is wrong. You are asked to prove that f(x)= 0 for all x in [a,b]. The "contradiction" of that is NOT that f(x) is non 0 for all x in [a,b], it is that f(x) is not 0 for at least one x1 in [a,b]. Since f(x) is continuous, there exist some [itex]\delta[/itex] such that f(x) is non-zero (and |f(x)|> 0) for all x in [itex](x_1-\delta,x_1+\delta). Then you know the integral on [itex](x_1-\delta,x_1+\delta) is non-zero.
  4. Jul 10, 2008 #3
    Ohhh I see what you mean :)
    thankss, i understand it better now
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