temperature gauge broken ?

Yesterday (it was cold) I noticed the temperature gauge only went up about 1/4 way from 'C' to 'H', and thought "oh crap, I got a thermostat problem". But when I started the car hours later to leave work, it was still at that position, and stayed there for the drive home. The next day when I drove to work it was an at about 1/8 way and didn't move. Clearly a malfunction, I guess either of the gauge itself or of the sensor.

I'd appreciate advice on this, for example, which is more likely to be the problem (sensor or gauge), or is it something else, what's the best way to diagnose it, etc. Thanks !