The parts pictured below are called balance staffs. The balance wheel, roller, and hairspring are carried by this part. These measure only about 3 to 5 mm in length!
Take a look at the two balance staffs in the photo above. The one on the left is the new part, and the one on the right is the old part. The top pivot is worn to a nub, and the bottom pivot is worn down to a sharp point.
This type of wear is a sign of a poorly maintained watch. The oil has dried up and contaminants have gotten into the jewel setting, causing this progressive damage. The watch continues to run as this wearthrough occurs.
Many people assume that a watch that is running does not need service. If you have such a watch that you value, or whose service history is unknown, it would be wise to have it inspected and serviced by a competent watchmaker.
Not only will your watch run better, but it will have a longer life.