Depends on the situation, whether it's worth getting worked up about. Generally, loyalty ranks high in my criteria of being/having a friend. If the bond is broken over something small, it would lead one to consider their value as a friend when the consequences were dire. Don't lie/don't straddle the fence. Shoot from the hip, and when you do, make sure it's dead on target!
I probably wouldn't take the chance and cut off all connections with them. I mean, if you're going to pretend like you're something else in front of my face, then go off and change your whole identity behind my back, than you might as well stay out of my life.