Emperor
The first Austrian Emperor was the last Holy Roman Emperor Francis II. In the face of aggressions by Napoleon, Francis feared for the future of the Holy Roman Empire. He wished to maintain his and his family's Imperial status in the event that the Holy Roman Empire should be dissolved, as it indeed was in 1806 when an Austrian-led army suffered a humiliating defeat at the Battle of Austerlitz. After which, the victorious Napoleon proceeded to dismantle the old "Reich" by severing a good portion from the empire and turning it into a separate Confederation of the Rhine. With the size of his imperial realm significantly reduced, Francis II, "Holy Roman Emperor" became Francis I, "Emperor of Austria". The new imperial title may have sounded less prestigious than the old one, but Francis' dynasty continued to rule from Austria and a Habsburg monarch was still an emperor ("Kaiser"), and not just merely a king ("König"), in name.
Western Roman Empire
The authority of Julius Nepos as Emperor was accepted not only by Odoacer in Italy, but by the Eastern Empire and Syagrius in Gaul (who had not recognized Romulus Augustulus). Nepos was murdered by his own soldiers in 480, a plot some attribute to Odoacer or the previous, deposed emperor Glycerius, and the Eastern Emperor Zeno chose not to appoint a new western emperor. Zeno, recognizing that no true Roman control remained over the territories legally governed by the Western court, instead chose to abolish the juridical division of the position of Emperor and declared himself the sole emperor of the Roman Empire. Zeno became the first sole Roman emperor since the division after Theodosius I, 85 years prior, and the position would never again be divided. As such, the (eastern) Roman emperors after 480 are the successors of the western ones, albeit only in a juridical sense. These emperors would continue to rule the Roman Empire until the Fall of Constantinople in 1453, nearly a thousand years later. As 480 marks the end of the juridical division of the empire into two separate imperial courts, some historians refer to the death of Nepos and abolition of the Western Empire by Zeno as the end of the Western Roman Empire.