I know this question was asked here before but I didn't get a straight answer. Some say Rome fell in 476 when Western Rome fell or 1453 when Eastern Rome (Byzantine Empire) fell. Yes, the fall of Western Rome was a major turning point in history. But was it the end of Rome? Some think that the Byzantine Empire continuation of the Rome of the classical era.
My History Professor gave us this example in class: Imagine if half of the US was conquered and Washington DC was sacked and occupied. Many people living in the remaining half of the US would consider it the conquered territory still part of the country just under enemy occupation.
So was the Byzantine Empire still Rome or a separate entity?
Short version: depends on how you think of it and who you ask. There's no objective "right" answer here; it's really a matter of "how much" (and, to some extent, "when".)
Now, I should point out that the inhabitants of Constantinople referred to themselves as "Rhomaioi", and everyone around them, allies and enemies alike, called them "Romans". That they traced their legitimacy in a more-or-less unbroken chain from the Eastern Roman Emperors was undisputed; so was the fact that their bureaucracy and tax system were inherited from Rome, and allowed the Emperor to marshal resources that other monarchs could only dream of. Until the loss of Egypt to the Arab invasions, Constantinople maintained an urban grain dole fed by Alexandrian grain ships, which was more or less lineally descended from the old Roman dole, and much of its urban culture came from Rome as well. The fact that the capital city wasn't well, Rome didn't matter so much; after all, the Western Roman court had decamped to Ravenna in AD 402, and before that, it had been in Milan.
But. It's pretty fair to argue that the Greek-speaking polity ruled by Basileus Basil II circa 1000AD is pretty different from the more Latin-speaking polity ruled by Caesar Justinian I from 527-565, and that in turn is different from the Europe-spanning Dominate welded back together by Diocletian in the late 200s, and that you have to draw the line to mark the differences somewhere. 476AD just happens to be a convenient point for that, with the extinction of one line of Roman emperors. Just so long as you keep in mind that for about a century afterwards, Eastern Roman Emperors would make attempts to bring the Western territories back under Imperial rule, and that it was through a mixture of happenstance and geopolitics--all which could've gone differently--that they didn't succeed.