Do you count the Byzantine as separate or the same as Rome?

Your talking about structures comprising huge numbers of people across multiple generations. There is no clear "death". Just the gradual shifting from one set of conditions to another. Pick any line in the sand, declare it to be the "end" of an empire, and you'll still find people living under its rules, speaking the language, and using the currency well afterward.

Hell, look at Britain. No longer the globe-strangling power that they were, but it's still the same country with the same rules and government and money.