When the thermostat is set on "auto" it turns off when the desired temperature is met, but then turns back on almost immediately. It turned on and off 40 times in one minute. If I leave it on "ON" there doesn't seem to be a problem. I live in an apartment so should I have someone fix or replace it, or is this how it should function?

I am not an expert in HVAC, but I have never experienced this problem before. I have rented many apartments in the past. The device seems to be malfunctioning. Notify you landlord or complex manager as soon as possible. You do not want the problem to persist and escalate into complete failure of the device. It's too cold to go without heat; well at least in Michigan it is.