Can somebody continue a road after it has been cut off/separated from its nearest city?

Question

• I have - - - - - o (a settlement at the end of my road).
• Now an opponent builds -- x ---o (a settlement along my road).
• Can I do o - - x - - - - o (build a second settlement at the other end), or do I lose my road once it is cut off from my original settlement?

Answer 1

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.

Answer 2

Yes.

The roads remain yours, and you can continue to expand them and settle on them.

There are two weird related rules, though:

1. An opponent's settlement at the end of your road prevents you from expanding through that settlement.

   o - - -x [cannot place more roads here]
   
   o - - -x- [can place more roads/settlements here]
   

   This is illustrated on the official website.

2. Even though you can keep building, your road is considered to be cut in half for the purposes of Longest Road.

   Longest Road length = 6
   o - - - - - -
   
   Longest Road length = 4
   o - - - -x- -
   

Answer 3

Yes -- the only requirement to build a settlement or road is that it be connected to an existing single road. That road doesn't need to be connected to anything else.