The Byzantine rulers thought of themselves as Roman emperors. How much was Byzantine civilization a continuation of Roman civilization and and to what extent was it something different? Why? Give an example of cultural continuity between Byzantium and Rome and one example of a discontinuity.
One discontinuity (although early on it was a continuity) was the development of a unique Byzantine form of architecture that used Near East motifs and forms as well as those of the Greeks. Early Byzantine architecture attempted to mimic that of Rome, but by around 700, it had developed into a unique style of its own, which both borrowed from and contributed to architecture in the Islamic world.
The greatest point of continuity was surely Roman Law. One of the greatest achievement in human history is the creation of the Roman Law Codes. It started early in Rome and continued to the Byzantine Empire. The earliest law code we have is the 12 Tables and the culmination of all of this comes from Justinian. A point of discontinuity is geography. Byzantium had it base of influence in the East instead of the West. Modern day Turkey became more and more important for Byzantium.
The Byzantine Empire was originally the eastern seat of the Roman Empire at Nicomedia until the seat was moved by Emperor Constantine to Byzantium, later Constantinople. One of the differences, or discontinuities, was that the Byzantium Empire outlived the classic Roman Empire, although after the fall of Rome around 476 AD, the Roman Empire was continued from the eastern seat in Byzantium/Constantinople. One continuity was that Latin survived as a "living" language in Byzantium next to Greek, the dominant language of the eastern empire.
To pick up on #4, one of the most important aspects when we consider these two civilisations is the way in which they developed their own distinctive "brand" or type of Christianity, and these two different types of Christianity persist unto this day in very different forms. The extent which Orthodox Christianity (emerging from the Byzantine empire) is different from the Christianity that emerged from the Roman empire is shown by the way in which there are immense theological differences and cultural aspects to the two distinctive types.
The one thing that always comes to mind for me, regarding these two civilizations, is the fact that the Byzantines considered themselves Romans. The term Byzantine did not come about until the eighteenth century. Outside of that, both posters above mentioned the differences in language and religious beliefs.
One example that might count both as continuity and discontinuity at the same time involves the Christianity eventually associated with each city. Both cities were nominally Christian, but each developed a distinctive kind of Christianity, and the difference persists to this day.
The major discontinuity between the two of these in my mind is language. The Byzantines, of course, spoke Greek. The Western Empire spoke Latin. The two also had different religions with Orthodoxy prevailing in the East and Catholicism in the West.
Byzantium was named the capitol in the East because the whole of the Roman Empire had gotten too big to administer effectively, or at least that's the reason most historians give. For quite some time there were dual capitols, but only Byzantium remained as a cultural and political force as it was able to withstand the barbarian attacks. Rome certainly imposed its political order on the East, and such institutions, like the Senate, continued to exist for several centuries after the western fall. Certainly the decline of trade in the West was never experienced in the East; perhaps that was one reason the Eastern "Empire" continued to survive until the 1400's.