Yes in general aircraft can depart with an inoperative (INOP) APU's. In flight the APU generally serves as a backup in the case of a failure of other systems (e.g. electrical failure). Prior to departure the pilots will check the aircraft "tech log" (aircraft technical logbook) which will indicate if the APU is not working. The pilots will then reference the MEL (Minimum Equipment List) which detail if there are any restrictions on departing without the APU - for example, often for ETOPS (Extended Twin Engine Operations) the APU is required to be serviceable.