It's a date, but not a terribly convincing one in my view. As far as imperial politics went Rome had not mattered for some time. For decades emperors had been accustomed to live elsewhere, like the easily defended town of Ravenna. Strictly-speaking Romulus Augustulus may not have been the last emperor. There existed another, Nepos, who had ruled at Rome before Augustulus and been obliged to move to Dalmatia which he still ruled with the support of the Eastern Roman Empire. Augustulus himself had no power and less respect. Even his name was a mockery of the sonorous title his father Orestes gave him. Orestes had renamed his son Augustus. The populace nicknamed him Augustulus, or 'Augustiekins'.
After he was quietly removed (his father having been murdered by Odoacer) there remained at least two imperial territories under Roman control: Dalmatia under Nepos, fated to last until occupied by the Ostrogoths in 480, and northern Gaul under Afranius Syagrius.
Syagrius was the son of Aegidius, the last Commander-in-Chief of Gaul, elevated to the post by the emperor Majorian. After Majorian's death the corridor connecting northern Gaul to Italy was cut by the Visigoths. Aegidius and Syagrius after him were isolated from the other remains of the Empire and became in effect independent rulers. What is important to note however is that they always considered themselves imperial governors, and ruled their territory in the name of Rome. This gave them legitimacy and enabled them to maintain their authority over the four provinces they controlled: Lugdunensis II, Lugdunensis III, Lugdunensis IV Senonia and Belgica II. It was this territory that became, in effect, the last remaining stretch of the Western Empire after the fall of Dalmatia. It would endure until 496/7, when it finally accepted the rule of Clovis, king of the Salian Franks.
Here is a more detailed history of the period. The map below shows the extent of the territory of Aegidius and Syagrius.It was about the size of Ireland, and contained some of the wealthiest lands of Gaul. It would later become part of the economic heartland of Mediaeval Europe. But more about all that in a future post.