Oh wow. Apparently, I've been playing Catan incorrectly the whole time. I never knew an opposing player's settlement can obstruct my own path (and split a path into two).
I'm too lazy to look it up right now, but I think you can continue to build on the road, BUT if they put a settlement at the end of your road you can't continue to build.
E.g. if [x] is your settlement, [o] is an opponent's and = is a road:
[x]==[0]= <-- you can build another road here
[x]==[0] <-- you can't build another road