In western schools is usually taught that the roman empire fell in 476 AD, and that his oriental part survived as “byzantine empire”. This definition is wrong for sure, for many reasons. According to history, the name “byzantine” was used for the first time around 1600 and was used many times during the enlightenment, in a derogatory sense.
The Enlightenment thinkers, in fact, valued the reason as the most important thing, and tought that the Medieval Period was the dark period of human intellect, but they also loved the Classical Antiquity. So, simply speaking, they invented a way to break the continuity to the Roman Empire into a dark period.
One of the most important examples is in the famous book “decline and fall of the roman empire” by Gibbon, how he uses insults and invectives, sometimes with a comical language, about the medieval empire, defining it as “kingdom of the Greek slaves and their historical servants” and a not interesting part of history.
If we don’t consider the name “byzantine”, we see that medieval romans simply called the themself, “romans” or “Greek speaking romans”. “Greek” was considered as a sort of insult and was used by western people in the same manners the Romans called them “barbarians” or “Latins”. By the contrary, eastern enemies of the Romans, form Arabs to Turks, called the heirs of Rome with the name of “Romans” and they tried to take their legacy every time they conquered the imperial territory. For example, the Seljuks founded the “Sultanate of Rum (Romans)”.
The roman society was not immutable and the empire in 1000 AD was really different from 200 AD, as it was very different from the Roman Republic and so on. During the first centuries, the medieval roman empire was the perfect continuation of the late roman empire, and after some time it simply adapted to the new challenges it faced.
Other people said that the medieval roman empire was no more empire because the capital was no longer Rome and the official language was greek. About the capital, we must remember that the Romans had many capitals during time, for example: Ravenna, Milan, Constantinople, Nicaea, Nicomedia, Antioch, Sirmium and others, so what they said is nosense.
About the language, we have to consider that the Greek and the roman world was symbiotic and the Greek was, since the early time, the idiom used by scholars and, in most cases, merchants.
In any case, using Greek as the official language, didn’t ended the use of Latin as it didn’t erased other languages used around the empire, like Coptic, Armaic, Illyric, Armenian and many languages from Caucasus and Anatolia.
The empire became more eastern and moved his center to the east during time, but this is not a reason to think that it simply vanished.