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    Real Analysis: Criteria for Integrability

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    Suppose the continuous function f:[a,b]->R has the property that:

    The integral from c to d f<=0 whenever a<=c<d<=b

    Prove that f(x)<=0 for all x in [a,b]. Is this true if we require only integrability of the function?

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    Solution Preview

    Suppose that f(x)>0 for some k such that a<=k<=b

    Then by laws of continuity

    lim h-->0 f(k+h)>0 and lim h->0 f(k-h)>0

    This means that we can always find an ...

    Solution Summary

    Integrability isused to prove a functional inequality.