Hmm, this requires an unambiguous definition of "continuity". What is it? late Byzantine Empire and Roman Republic differed far too greatly. What defines a continuing institution? One may say the government, the fact that the Byzantine considered themselves Romans. Hence the name of Byzantium - Nova Roma. Constantinople was the unofficial, popular name.
That is splendid, but what shall we make of Charlemagne's Empire? Or that of Otto I? What is the difference between Holy Roman Empire and that of the Byzantines? Both considered themselves as the continuation of Rome.
Or if you truly wish to make a farcical argument, then why not consider the Tsardom of Russia? After all, Ivan III, the Grand Duke of Muscovy married Sophia Paleologue. And she was the niece of the last Byzantine Emperor, Konstantinos XI Palaiologo. This and the fact that Moskva was the only remaining major (the Southern Slavs in the Balkans were subdued by the Ottomans) Eastern Orthodox capital made Kremlin adopt the "Third Rome" doctrine. Basically, Russia was the heir of Rome, even though it was in a rather more religious context. "Tsar" entered the Russian usage, derived from "Caesar".
Next, Mehmed II, the vanquisher of the Byzantines, was likewise not immune to self-flattery. After his momentous conquest, he self-styled himself as a "Caesar" (Kyiser) of the Romans (Rum). Evidence aplenty exists that he did aim to conquer Rome itself, but then he was killed. It is regrettable that the subsequent Sultans did not care for the significance of their past conquest as much, or to be accurate, at all.
All throughout history numerous figures and nations have claimed themselves the continuation of Rome, so a definition is needed. Otherwise, Russia has not too shabby of a claim that it was the third manifestation, third continuation of the fabled Roman Empire. That is obviously a nationalistic, pompous pretence, but without a valid definition of what constitutes as "continuation of an empire", the claim is as good as any.
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