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Math Help - One Question In Real Analysis ..

  1. #1
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    One Question In Real Analysis ..

    Hi all..

    i want to ask you about this question in real analysis..

    "suppose that f continuous on [a,b] , that f(x) >= 0 (more or equal 0 ) for all x in [a,b] and that =0

    prove that f(x)=0 for all x in [a,b] "
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  2. #2
    MHF Contributor Bruno J.'s Avatar
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    Suppose \exists c \in [a,b] such that f(c) > 0. Then there exists a neighborhood N of c such that f(x)>f(c)/2 for all x \in N \cap [a,b]. (Prove this using the continuity of f at c.) Then 0 = \int_a^b f \geq \int_{N \cap [a,b]} f(c)/2 > 0 which is false.
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  3. #3
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    thanx for your answer..
    but i want the solution in another way..
    can i solve it by cauchy criteria and tagged partition or squeezing theorem..
    im Really Confused and i cant join the information together..
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  4. #4
    MHF Contributor Bruno J.'s Avatar
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    I don't see how anything that you mention really has much to do with the problem. The solution I gave is quite standard : suppose the function is not identically zero, then it's bounded away from zero on a small subinterval, and hence its integral on this small subinterval is positive, and the integral on the whole interval is at least the integral on this small subinterval.
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