The byzantine empire was the only classical empire that survived into the postclassical period. why did it survive when all the others did not?

the Byzantines were able to maintain their main areas. There were no attacks by tribal or pastoral people on Anatolia or Egypt, and just a few in the Black Sea region. For the most part the Byzantine Empire was not an expansionist empire, and this gave it more stability.  

The tax system and the bureaucracy were already in place, and the bureaucracy acted as a check on the nobility. It was good way for them to earn reputation by serving in the government. Under Justinian, a law code was created. It came directly from the emperor, establishing him as the ultimate authority on justice. This was an important check on the power of the nobility as well , helping to keep them from growing powerful enough to attempt to take over the central government. It gave the imperial office a lot of authority and prestige.  

Outside of Constantinople we see evidence of a strong agricultural economy. Although there were many large urban areas, the vast majority of the empire was made up of agricultural villages. They were the primary tax source and they supplied grain and agricultural products to the urban populations. At the same time, the Byzantines were the best economy in the region. They were the major trading power in the Mediterranean. They produced goods like textiles (especially silk), jewelry, and other crafts, but they also acted as the middleman for trade between east and west.

A growing cultural identity and social unity. The elements of this unity included, first and foremost, a long-lasting Roman identity. Byzantines believed they were keeping in the glorious Roman tradition, and they always presented themselves this way. Constantinople was also an important item of Byzantine unity. It was seen as the greatest city in the Mediterranean world and had overcame dozens of attacks. It was also portrayed as a whole Christian city, even if this wasn't actually true. There were lots of different kinds of Christians, plus Jews and later Muslims.