According to the Byzantium's, they are the old Roman Empire (and truthfully, they basically were. The Roman Empire was split into 4 parts eventually, with 2-4 emperors over it. By the time Rome fell it was seen more as the ghetto part of the Empire to the point that even the Capital had been changed from Rome to Constantinople).
In that light, if Rome was founded in 753 BC, and Constantinople fell in 1204 (yes, it didn't officially fall until 1453, but the "Roman" part of it fell in 1204 to the 4th Crusade) that means it lasted almost 2000 years as the Roman Empire/State.
Though Europe called it Byzantine, the Byzantines called themselves Romans.
They survived losing Constantinople.
The classical empire essentially died 6th century iirc. Well final death some forms of the Roman administration survived into the 7th.
Or in modern terms say USA created another capital in say LA. Washington is lost over the next 1000 years most if what's left is lost and then the lose LA but the capital is moved San Francisco.
50 years latter they recapture LA move capital back. Then 200 years later they lose LA and a few remnants last a decade or so.
There's a political continuation from Augustus through to Constantine XI.