Yes, there were discontinuities. But there were also continuities. Among the latter was the name the people gave to their country - The Roman Empire - which was in continuous use from 27 BC to AD 1453.
Its interesting that you brought up China. A better comparison is that between the transition from Diocletian (r 284-305) to the Heraclian dynasty (r 610-711) in the Roman Empire, and the transition from the Han dynasty (r 206 BC-AD 220) to the Tang dynasty (r 618-907) in the Chinese Empire. After all, post-Sui China was nowhere refered to as "the Luoyang Empire". It remained China.
However, I do agree with an earlier post, that one can refer to the Byzantine phase, or Byzantine Period (c 610-1453), of the Eastern Roman Empire (395-1453), though this must be understood as an artificial division, and simply for the practical purpose of dividing-up the otherwise massive history of the Roman Empire (27 BC-AD 1453), and even bigger history of the Roman state in all its incarnations (753 BC-AD 1453).