Heat pump Not turning off
Depending on the exact details of your heat pump, it may be functioning exactly as designed.

Modern efficient heat pumps are commonly variable speed (may say "inverter drive" or something like that), so that the compressor speed can be matched to the load 


more load, more speed, more power
less load, less speed, less power.


So, rather than overshooting and turning off until the temperature falls, they simply slow down to match the output to the load and keep running, maintaining the temperature as set.

An additional factor with cold climate air-air heat pumps in cold weather is that they occasionally need to defrost the outside coils.