In their minds, they were the Roman Empire. The notion of a Byzantine Empire is a much more modern construction, mostly from Western scholars who had a hard time viewing Greek speaking Orthodox Christians as the successors of Rome. But the Emperor in Constantinople would have traced his power back to Augustus, and if you wandered the countryside and asked someone what country you were in, you would have been told "Rome", possibly with an odd look.

The notion of being Roman had long since stopped being linked to being from a specific city on the banks of the Tiber. There were some differences. The capital was in Constantinople, but it had been moved there by Constantine, who was unquestionably a Roman emperor. The language was Greek, but the eastern parts of the empire had been Greek speaking even in the time of Augustus. (Latin did remain the official court language there for over a century after Rome itself fell). The religion was different, but hadn't been until 1054.

So as far as they were concerned, there was no discontinuity at all. The Roman Empire lived on until 1453 when The City fell.