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!
Interesting. Thanks for the big picture.
However, if agriculture and trade and so forth continued uninterrupted through all that time, what caused the tremendous decline in population and economic activity in Europe in the period from the 6th century to the 12th, roughly? What happened to Roman Britain, Roman Gaul, Roman North Africa, Roman Dalmatia, etc., if there was all this continuity?
Interesting!