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Recently I have come across a proof with application of characteristic function.

After some steps in the proof, it concluded that there is a neighborhood of 0 such that the characteristic function is constant at 1, then it said the characteristic function is constant at 1 everywhere over the domain.

I suspect that "If there exists a neighborhood of 0 such that the characteristic function is constant, it is constant everywhere." Is this correct?

I have tried to search from the web regarding this but found nothing. Would anyone suggest me some good reference on characteristic function as well.

Thanks.

Wayne

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# Characteristic function

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