Can somebody continue a road after it has been cut off/separated from its nearest city?
To build a settlement, you just need to obey the relevant rules for building a settlement, which are:


  You may only build a settlement at an intersection if all 3 of the
  adjacent intersections are vacant (i.e., none are occupied by any
  settlements or cities—even yours).
  
  Each of your settlements must connect to at least 1 of your own roads.


Assuming that the site you want to build at isn't adjacent to another settlement or city, then all you need to worry about is that you have a connecting road.  And in this instance you do: there's no reason to suppose a road ceases to be yours just because someone has interrupted its length.