Did a quick search and found a few answers to help get you started:

• Did "Italians" in the early middle ages still think of themselves as Roman?

This one might be exactly what you're looking for, but here are two more that touch on the topic:

• At what point was it obvious to Italians that they were no longer Roman?

• What effect did the fall of the Western Roman empire have on the population in Italy?

Hope this helps! As always, further discussion and new answers are welcome and appreciated.

EDIT: Had an extra word in there

Did People in Britain ever really consider themselves Roman in the first place, besides the ones who actually are Roman in origin?

I always got the impression that the people of the Roman Empire never really developed a cohesive, unified national identity, at least not to the extent that, say, the Han Chinese did.