In Catan, when a settlement breaks a road. Do the other existing roads sit there? or are they removed?
Looking at http://www.catan.com/service/game-rules I can't find the word 'displace' so i'm not sure exactly what rule you are referring to. So I will answer your question by explaining what would happen quoting the rules.

Building a settlement.


  You must meet 2 conditions when building a settlement: (1) Your
  settlement must always connect to 1 or more of your own roads. (2) You
  must observe the Distance Rule.


There is nothing there about removing roads from the map.

in the Longest Road section it states.  


  You can break an opponent’s road by building a settlement on an
  unoccupied intersection along his road!


This might apply in the example you have given.  Your longest road length will no be either section A or B, depending which is longer.  There is nothing in the rules about removing roads from the map here either.