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Mean value theorem for integration fails for vector-valued functions

If the function G {\displaystyle G} returns a multi-dimensional vector, then the MVT for integration is not true, even if the domain of G {\displaystyle G} is also multi-dimensional.

For example, consider the following 2-dimensional function defined on an n {\displaystyle n} -dimensional cube:

Then, by symmetry it is easy to see that the mean value of G {\displaystyle G} over its domain is (0,0):

However, there is no point in which G = (0, 0) {\displaystyle G=(0,0)}, because | G | = 1 {\displaystyle |G|=1} everywhere.