After establishing himself as the Emperor of the Western Roman Empire, the Roman Emperor Constantine moved the capital of the Roman Empire to Byzantium. The Empire itself didn't cease to being the Roman Empire at that time and magically transform into the Byzantine Empire. The lines of Emperors that ruled from Byzantium (Constantinopolis), whether hereditary or no, were still the Emperors of the Roman Empire. The same Roman Empire of Augustus Caesar.
Even after the fall of the city of Rome itself to the barbarian invasions, the eastern portion of the Empire was still the Roman Empire, just minus the western parts from earlier centuries.
Constantine was not the first Byzantine emperor, nor the last Roman emperor, he was just another Roman emperor and there were a good 1000 years of Roman Emperor's that followed him afterwards, irregardless of the fall of the city of Rome and it's inclusion in the Roman Empire.
Seems pretty cut and dry to me really, I don't get what the fuss is about. Clearly , what history refers to as the Byzantine Empire is just another name for what used to be referred to as the Roman Empire. Regardless of the Language differences they had, the differences in culture, art , architecture and religion.
At the founding of the Roman Empire, they were all polytheistic worshippers religiously, later emperor and polytheistic worshippers and after that Christians....yet all of these peoples, and the Empire that contained them throughout its history , were ROMAN.
People, languages, cultures all change over that much time, still, despite all of the vast changes over said period, the citizens of the large empire of the Meditteranean were ROMAN, even after the city itself fell into outside hands.