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Math Help - intergration

  1. #1
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    intergration

    suppose f:[a,b]-> real numbers is continuos and for all continuous functions g:[a,b] -> real numbers, then intergrate from b to a of (f.g)=0. show that f is identically 0.

    how do i go about solving this?
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  2. #2
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    Does f.g means f multiplied by g?

    You can pick any g, so let g = f
    Now what does the statement simplify to?
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  3. #3
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    ya it means reply.
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  4. #4
    MHF Contributor Drexel28's Avatar
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    Quote Originally Posted by alexandrabel90 View Post
    suppose f:[a,b]-> real numbers is continuos and for all continuous functions g:[a,b] -> real numbers, then intergrate from b to a of (f.g)=0. show that f is identically 0.

    how do i go about solving this?
    Just as a remark, this is true more generally in the sense that if \displaystyle \int_a^b f(x)h(x)\text{ }dx=0 for every h \in C^k[a,b] (for some arbitrary but fixed k\geqslant 0) with h(a)=h(b)=0 then \displaystyle f\equiv 0. The proof is not much different, merely take h(x)=f(x)(x-a)(x-b) to conclude. I mention this because it shows that C^k[a,b] separates points with respect to the usual weak topology on it. In this context this theorem (as simple as it is) actually has a name. It's the Fundamental Lemma of the Calculus of Variations.
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