I just learned that the people of the Byzantine Empire still called themselves as Roman but in comparison to the Western Roman Empire, the dominant language was Greek which may have been mostly influenced by the Greek culture that was passed down through generations after Alexander the Great conquered most of the East.
I also learned that the language that one speaks influences a lot in one's culture so if the people in the Byzantine Empire were speaking in Greek then this may also mean that their culture was also influenced by that of Greek descent or descent from the Hellenistic Era.
So I wish to ask that since the Byzantine Empire was originally a part of the Roman Empire and was still a part of the Roman Empire until it fell to the hands of the Ottomans, how long and how much of the Roman influence stayed in the Byzantine Empire?
This is a bit tricky to answer as culture is never static and is always developing and evolving as time goes on and situations change, as well as factoring in the already established cultures of the regions annexed by Rome, especially in the case of Greece which already had a hard established culture developed during the classical days of the city-states. The Roman Empire itself was incredibly diverse in culture, which goes with the territory of annexing a lot of land not populated with Latin Italians.
Roman culture, just like any culture throughout history, was fluid and always changing, and was never uniform throughout the entire history from the Republic to the Empire. Roman customs and culture during the late empire, say the 4th Century, was drastically different from the culture and customs they had during the Punic Wars of the 3rd Century BC.
For a few examples, the Romans shifted religion from worshiping Jove (Jupiter) and the rest of the Roman pantheon of gods to shifting religion to revolve around Sol Invictus (the Sun God) by the 2nd and 3rd Century AD, and later abandoning Paganism altogether for Christianity. Also, the ever iconic gladiatorial games began to fall out of popularity in the 2nd Century AD and were abolished altogether in the 4th Century with the rise of Christianity. These two examples are just to show that Roman culture wasn't static, and doesn't even account for the lands that Rome conquered that already had an established, pervasive culture such as Hellenized Greece, Egypt, and the Levant, which largely held onto their pre-roman traditions throughout their time in the Empire.
In the case of the Eastern Roman Empire after the fall of the West, the state carried on with Latin and the language of government and administration, but the common population still spoke their respective native tongues, with the large majority still speaking Greek, as centuries went on, the empire continued to change just as roman culture changed during the days of the Republic and early Empire, the language of the Eastern Empire's administration was changed to Greek from Latin during the reign of Heraclius in the early 600's, and the early Chalcedonian Christian practices slowly evolved to become what we know today as Eastern Orthodox, and so on. The Eastern Roman Empire didn't discard Roman culture and practices so much as they built on it as time passed.
Reversely, there were aspects of the old Rome that the Byzantines held onto to the very day they fell to the Ottomans. They still followed Roman law, which received a condensed codification in Justinian's Corpus Juris Civilis. The Byzantines were also rabid fans of chariot racing in the same fashion that was played out in the Circus Maximus in Rome, and the races continued until the state could no longer provide the funds to host the games, with records of the bigger races fading out around the time Constantinople was sacked by the Crusaders in 1204.
To sum it up, the Byzantine Empire still retained many of their Roman roots in the forms of sport/entertainment and government to the end, while at the same time building on old traditions of the Romans and classical Greeks to make something that we see as unique to the Byzantine Empire. To the contemporaries of the time, however, the state and its people was still seen as the Roman Empire by surrounding nations such as the Seljuk/Ottoman Turks to the east, and the Bulgarians, Serbians, Germans and Hungarians to the West.
If you want further reading into the culture of the Byzantine Empire and it's unique blend of Greek and Roman traditions, here's a worthwhile read:
Mango, Cyril A. (2007). Byzantium: The Empire of the New Rome. Athens: Educational Institution of the National Bank of Greece.
There should be a translation by Dimitris Tsoungarakis as the original text is Greek.