Tiberius, as Red Peasant puts it, change is nothing to frown upon. It's a natural evolution of things. The Roman Empire was barely Roman to begin with. I won't go into a "when did the Roman imperium start" type of debate, as I do view (as RP) the early imperium as an empire practically. I'd say that ever since the second punic war Rome was an empire alright. But for simplicity's sake I've picked the "official" starting date of the imperium. But ever since the Romans began absorbing non Romans, they were barely Roman/Italian. There was very early dynasties of non-Italian emperors, the culture of the empire was anyway never Roman (graeco-roman and later greaco-roman-christian would describe it more accurately) so that's not an argument contra the continuation of the eastern empire.

True, the latter evolved (or devolved, any way you like it) and became something completely different. But an empire is not the people that consist it (to put it in perspective the Hapsburgian empire was the same, the Hapsburgian empire, never mind if it was centered in Italy, Spain, Germany or Austria - true, it was a single dynasty, but that's a step back from the Roman concept of an empire, no?).

The Roman entity was discontinued in the West, when the "barbarians" took over, but it kept going in the East. True, the majority of the population spoke Greek and bore a Greek culture, but it was always like that in that part of the world. Most of the "Roman" writers, artists and administrators in the late imperial times, were nothing like Romans - Greek, Gauls, Illyrians and many others, even Goths manned the administration and army for the greater part, the "true Romans" were already a small minority in "their" state.

The eastern Romans, although most tend to call them Greeks, didn't call themselves like that, only to designate their particular ethnic group (and that only after the term "Greek" or "Hellen" correctly, seized to be badmouthing - it equalled "pagan", you know). They called themselves Romans, "Romeoi" (that's Greek for "Romans") or more commonly Romioi. The separation of the Roman empire from the "Byzantine" empire is artificial and invalid - it's the same entity. It had changed over time, that's true... but what doesn't? We ain't talking 100 or 200 years here, we are talking 1.500 + years (if you accept RPs notion, 1700 years).

But if you really want to go down that line... here's some contra argumentation: When did the RE seize to be RE? Is it about Rome? Well, the capital had moved from Rome even before the fall of the West. Is it about the language? Well, Latin kept on being used until 1204 to the least, although only in a narrow part of the whole structure. But even that way, up until the times of Heraclius, the Latin language was used solely as the official language of the empire. Most of the people spoke Greek, of course, but that was the case even in the Roman empire in the 1st century AD. Was it about the Romans and Italians as heads of the state? Well, then 2/3 of the emperors are not "Romans", huh?