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I have this question which asks for a 2*pi periodic function on the reals, that is integrable on [-pi, pi] but fails to satisfy the mean value theorem for integrals.

The question also says to help answer the above question, you may wish to show that the function:

g: [0,1] -> reals given by,

g(x) = sin(1/x) if x is non-zero

g(x) = 0 if x=0

Now I've shown that the above function is true, but I have no idea how to show that it fails to satisfy the mean value theorem or how it relates to a function that is 2*pi periodic and integrable on the given interval.

Please help!