Wasn't Rome sacked by Goths and Vandals, and then taken over completely by the Goths (Theodoric Amalung, and those guys) in the 5th century (give or take on the dates, my books are in the other room)?
I understand the greater continuity of Byzantium, but did that extend to regular trade as far as Britain?
Self ping
Technically that disrupted the continuity of government in the Western Empire, but it was still the same old place.
The West doesn't end until about 538 AD (there's a precise date for this) according to dendrochronology which shows a Fimbulwinter settling in for something like 3 to 5 years.
The Eastern Empire had unbroken continuity up until the city of Byzantium was sacked by the Crusaders, but then reasserted itself and lasted until the early 1400s when it was taken over by the Turks WHO RAN IT EXACTLY THE SAME WAY.
In fact, a good case can be made that the Eastern Empire didn't really end until the peace settlement of WWI that dismantled the Ottoman Empire and re-established the Arab states (which had been out of business for a thousand years).
The Roman Empire was a very large place ~ kind of like the US, and destruction in one quarter did not mean destruction in the others ~ no more than Katrina's destruction of New Orleans has any significance to the existence of Manhattan!