UNESCO's World Heritage listing for Ancient Jericho/Tell es-Sultan describes traces of human activity from around 10,500 BCE. By the ninth to eighth millennia BCE, a substantial permanent settlement had emerged with a wall, ditch, and tower, illustrating early social organization.
The archaeological sequence records multiple phases of occupation, which do not by themselves establish unbroken habitation across the entire period. The relation between the ancient tell and the modern city must also be distinguished from the question of when a city, rather than a settlement, began.
Damascus also has a very long urban history and is described by UNESCO as one of the world's oldest continually inhabited cities. This does not directly refute Jericho's claim. Prehistoric activity, permanent settlement, urban formation and uninterrupted habitation are different measures, so an early archaeological date alone cannot determine a unique winner.