Montana saw some brutally cold temperatures in December. How low did your thermometer go? Well, it probably wasn't low enough as it was too cold for most thermometers.
It was one of the coldest airmasses ever to hit Montana and temperatures may have even been colder than what your thermometer said. Elk Park north of Butte was one of the coldest locations checking in a 50 below. The Great Falls National Weather Service acknowledged that reading and further stated "the temperature sensor has stopped reporting, so likely it's at a minimum operating value". It was too cold for the instrument and that likely was the case for your thermometer too.
If your thermometer showed a ridiculous number, stopped at 40 below, or showed dashes, it likely malfunctioned. Most thermometers do not contain mercury anymore, but those would not have worked as mercury becomes a solid at 38 below.
Many alcohol and digital thermometers will only work down to -20. Digital instruments utilize electronics that read voltage from a component that changes resistance with temperature, requiring battery power. Unless the battery is lithium, most batteries will not work in extreme cold.
If you have a colored ethanol thermometer, which is unlikely, then it may have been accurate because those work down to below -40 all the way down to 94 below zero.
It was one of the coldest airmasses ever for parts of Montana, and it might have been even colder than what most thermometers read.