What 2 important functions did byzantine empire serve

It helped in preserving roman and greek culture. justinian made laws called the Justinian Code which to this day, is the basic law code for many western countries.
the byzantine empire also helped in cultural diffusion.during the crusades of the 11th and 12th centuries, western europeans going to the holy land had to first pass through the byzantine empire. as a result they brought many of those ancient greek and roman accomplishments back to western europe. 

hope this helped!