A few things from the front section of our MEL---
--The Commander shall satisfy himself before the aircraft takes off that the flight may commence in compliance with the conditions of the MEL.
--MEL conditions and limitations do not relieve the Commander from determining that the aircraft is in a fit condition for safe operation with specific unserviceablities. If a failure occurs after the start of taxi and before the start of the take off roll, any decision to continue the flight shall be subject to the Commanders judgment and good airmanship.
--It is within the authority of the Commander to accept a defect without consulting a ground engineer after the doors are closed provided the defect does not require a maintenance (M) specific procedure and it is covered in the MEL.
So, yes you can continue under certain failures and yes you would write it up for the next station to action.
If for example you managed to reset the system and the failure cleared you would still write up the failure in the book detailing what happened. This is for trend monitoring etc