It depends on what these key words mean - "operate" on what level? and "fall" as in 476? A number of cities continued to provide public services, pay to keep up their existing infrastructure and construct new buildings. It depends on how you define the specific romanitas of a city! In Ostrogothic Italy, some aqueducts were repaired or built anew in Ravenna, Rome, Naples, for instance. Vandal North Africa also remained pretty urban. Overall, cities definitely get less important for a few hundred years, for sure.