The idea that the Roman empire extends to Byzantine times comes from Edward Gibbons' "The Decline and Fall of The Roman Empire" Ed had so much fun writing about the later roman empire that he ended up filling 6 volumes to tell us that the Byzantines were roman enough to let them carry the story into a much later era. Weather or not that's true is one of those arguments historians like to have. Go wiki Decline and Fall if you like, but I think Gibbons makes a pretty good argument. Either way, it's considered essential reading on the subject, especially since so many later historians start with his work, then clarify, expand, or correct his facts. If you happen to like stuffy, witless, english history books, you'll find no better read than this.