When did the Roman Empire stop being the Roman Empire

[Epictetus]

I think this article is quite good:

What, if anything, is a Byzantine?

It details what people of different regions and different times considered themselves.

Unfortunately it doesn't list sources, but it seems to answer the problem.

Edit:



Howard, I don't read Greek but supposedly these inscriptions state:

Romanos, emperor of the Romaioi
Eudokia, empress of the Romaioi

This is the so-called Romanos Ivory, or Christ Crowns Romanos and Eudokia. "Romaioi" is Greek for "Roman" so here is some proof that the "Byzantines" considered themselves "Roman."

The ivory is dated to either 945 - 949 or 1068 - 1079 and I got the information from Byzantine Empresses by Lynda Garland. The page about the ivory is available on Google Books.