Why shouldn’t we speak of continuity in case an entity has been split? Does continuity stops when a thing splits? Suppose a country is split into two separate countries (like the Czech Republic and Slovakia). Looking from the present to the past there is continuity back till the time that the original country was founded or came into being. So, looking backwards there is a continuity from the Czech Republic in 2007 back to the foundation of Czechoslovakia in 1918. But can we say that there is a continuity from Czechoslovakia to the Czech Republic? If we say no, how about Spain, if the Bask country would separate and would become an independent country? Then we should still say that there is a continuity of Spain from the times far back in history before the date of independence of the Bask state. If we say yes, how about my body and my toe? Is it important that my toe is only a little part of my body that we say that there is a physical continuity of my body from my birth on (at least)? But how big (in percentage of the combined mass of body and toe) must my toe be that we deny that there is a physical continuity of my body?
Etc. It looks like the problem of the ship of Theseus.