Answer:
The Byzantine Empire was the eastern continuation of the Roman Empire after the Western Roman Empire's fall in the fifth century CE.
The Byzantine Empire initially maintained many Roman systems of governance and law and aspects of Roman culture. The Byzantines called themselves "Roman". The term "Byzantine Empire" was not used until well after te fall of the Empire
people living under the Byzantine Empire continued to see themselves as Romans and continued to refer to their empire as the Roman Empire; the terms Byzantine Empire and Eastern Roman Empire were created much later.
Constantine legalized Christianity. However, this was still a period of transition. It wasn't until later, under Theodosius I—who ruled from 379 CE to 395 CE—that Christianity became the official state religion of the Roman Empire (both East and West).
Even though the Byzantine Empire is considered to start with Constantine's moving the capital to Byzantium, it was not considered a separate empire by historians until the fall of the Western Roman Empire in 476.
