When did ancient Rome fall?

The Byzantine part is an invention by historians as we all know that they did not call themselves. I consider the East after the supposed ending of the western half of Rome as more or less a continuation of the Roman empire. Even from Constantinople the Emperor and the entire court regarded themselves as Romans. To the Eastern Romans it was only they who could be called "Emperor of the Romans" not some semi illiterate 'barbarian' (as they veiwed it) like Charlemagne or anyone else endorsed by the Pope as the so called 'Holy Roman Emperor'.