"which explains one way in which the byzantine empire continued the cultural traditions of the romans?
by adapting some ancient greek ideas
by using latin as the primary written language
by restricting women to the inside of the home
by distracting the people with gladiatorial contests"

The option which explains one way in which the Byzantine Empire continued the cultural traditions of the Romans is the second one - by using Latin as the primary written language, at least until the 7th century when they started using Greek.
A is a weird option - how can adapting Greek ideas be a continuation of Roman traditions? C is insulting. D is incorrect as Byzantine Empire didn't have gladiators.