[Tuco Ramirez]

They didn't regard themselves as Romans; they regarded themselves as (and spoke) Greek.

[Destro]

Are you ignorant on purpose?

From the article:

In the provinces close to Constantinople, where the Greek language predominated over the Latin of Old Rome, the idea of Roman citizenship and identity appealed to a broad segment of the population. Greek speaking citizens were proud to be Romans: in Latin, "Romani," or, in Greek, "Romaioi." The word "Romaioi" became descriptive of the Greek speaking population of the Empire. The old ethnic name applied to Greeks, "Hellene",fell into disuse.

[StAthanasiustheGreat]

I would certainly consider the Eastern Empire, a continuation of the Roman Empire. Perhaps an accurate historical term would be the Roman Empire during the Byzantine Period. Besides, then we can say Rome lasted from 753 B.C. - A.D. 1476. Not a bad little empire.