In what ways did Byzantium serve as a bridge from the classical age to the Middle Ages in Europe?
In one sense, one could argue that Europe would not have reached the Middle Ages in its then-current form had not the Byzantine Empire kept the Persians and other groups bottled up to the east of Europe. As Lars Brownworth has argued--although overstating the case at times--the Byzantine Empire, or Eastern Roman Empire, which remained militarily intact long after the Western Roman Empire imploded under the weight of internal strife and external defeat in about 480 AD, kept Islam from over-running Europe while it was still recovering from Rome's disappearance as a military, economic, social, and religious force. It is reasonable to believe, for example, that David Gibbon was correct when he said in the Decline and Fall of the Roman Empire that, without the Byzantine Empire, the muezzin's call to prayer would have been heard in towers above Oxford (see Brownworth, Lost to the West). The Middle Ages, as we understand the term, might not have occurred.
The answer to the question, however, is much more complex. Because the Byzantine Empire survived until the early 1400's, it political and cultural traditions also survived well into what we now call the Middle Ages. The Byzantine Empire, partly because it looked to Greece for much of its heritage, spoke Greek (also, Latin, but Greek was preferred) and educated its citizens in a classically Greek manner: they learned the philosophy of Plato and Aristotle; rhetoric; mathematics; the literature of Homer and Aristophanes. In short, the Byzantine Empire largely modelled itself on Greece in the Classic Age and, despite influences from Europe, seemed closer to Greece and the East from the perspective of Western Europe.
The empire's power and longevity and "Greek-ness" (excuse the awkward word) all combined to form a cultural and, because the empire protected Europe from invasion, a physical, bridge between Greek classicism--in the slightly modified form of the Byzantines--and Europe. One can make a credible argument that the Byzantine Empire created a conduit by which Greek classic knowledge and culture became part of Europe's Middle Ages.