That is for the Eastern Roman Empire, which did obviously have the most legitimate claim to be the “successor” of the legacy of the Roman Empire (because they are a direct lineage of the Roman Empire).
But at that point there had been a thousand years of other empires also claiming to be the continuation of the Roman Empire. To the point where the Eastern Roman Empire is often more commonly referred to as the Byzantine Empire
The fall of the Western Roman Empire in 476 is generally more accepted as the fall of the Roman Empire. As the other comments say, that is when the city of Rome was sacked.
That is for the Eastern Roman Empire, which did obviously have the most legitimate claim to be the “successor” of the legacy of the Roman Empire (because they are a direct lineage of the Roman Empire).
But at that point there had been a thousand years of other empires also claiming to be the continuation of the Roman Empire. To the point where the Eastern Roman Empire is often more commonly referred to as the Byzantine Empire
The fall of the Western Roman Empire in 476 is generally more accepted as the fall of the Roman Empire. As the other comments say, that is when the city of Rome was sacked.