Broken key still working in lock - is the security of the lock compromised?

Hi it depends, if the lock is internal probably is two or three lever low security lock and if you mean the piece of the key is inside that there is a big space in the bottom of lock and cannot interfere to the lock performance,please specify the lock for me for accurate answer,hi again as I said 3 lever mortice locks in total are low secure the part of tension bit is there and the rest jiggling that's why is open plus keys have two identical side and for opening half key is enough but in one side of door depends which side is broken check if the key open the door in both side( back and front) if yes the lock have no security at all if no then no problem lock is OK become as you look at the other keys left and right are identical half is for front other half is for the back ,I hope you have got your answer.

if on the outside then yes, if on the inside could cause an issue when using another key the other side remove broke key or replace with a local locksmith.

No.

Internal doors don’t really require any more than 2 or 3 levers for security. Front doors require 5 lever locks British standard but 3 lever is ample for interior doors. As for the key itself, if it still works on both sides of the lock then that would be odd as a mortise key is symmetrical so it can operate the lock from both sides. Normally if a part has broken off the key and still works in a mortise lock it would be the part of the key that throws the bolt locked/unlocked and not the part of the key that is lifting the lever pack as the bitting part of the key needs to be accurate as there is as lot less tolerance. I would check to see that it still works on either side of the door before looking to replace it.