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    34.12 Show that if f is a continuous real-valued function on [a,b] satisfying
    ∫ f(x)g(x) dx = 0 for every continuous function g on [a,b], then f(x) = 0 for
    all x in [a,b].

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    Solution Preview

    Please see the attachment.

    We need to use the following properties of integrals. Suppose is integrable on the interval .
    P1: If for any , then
    P2: For any , we have
    P3: If for any , then .

    Suppose is a continuous function on and satisfies for every continuous ...

    Solution Summary

    This is a proof regarding a continuous real-valued function.