Russia in the 1990s, Haiti, Somali, large occasional power outages. Things go on… Cities are so large that they still somewhat go on functioning; education, hospitals, public transport, local shops. Because what is there to do otherwise? Small bands of armed bandits cause most of the problems, but that never last too long, killing each other. Small regions that break off suffer the most, as resources in those locations get scarce, they get overrun constantly with armed bandits. Small communities start to protect themselves, eventually successfully, and work to re-establish relations with the larger cities.
Most people that can. leave. Wars then go on based on funding from outside interests.