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Math Help - A question about Mean-Value Theorem for multiple integrals

  1. #1
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    A question about Mean-Value Theorem for multiple integrals

    In page 401 of Apostol's "Mathematical Analysis"(see figure below),

    the underlined statement seems to use the Intermediate value theorem to assure the existence of {\bf x}_0. But the Intermediate value theorem is applicable only when m and M are values of f for some point in S, which, however, can not be guaranteed by conditions of the theorem and the NOTE. If we additionally assume S is compact, the conclusion can hold. Whether or not there is a mistake here, or is there any other means to prove the existence of {\bf x}_0? Thanks!
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  2. #2
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    sorry for posting in a wrong subforum due to carelessness, please move to analysis subforum.
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