A question about Mean-Value Theorem for multiple integrals
In page 401 of Apostol's "Mathematical Analysis"(see figure below),
http://i50.tinypic.com/2cokfwp.jpg
the underlined statement seems to use the Intermediate value theorem to assure the existence of 
. 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 ? Thanks!
