This paper studies an unmanned aerial vehicle (UAV)-enabled downlink data transmission network for ground users in the presence of multiple no-fly zones (NFZs). To improved the efficiency of data transmission under fairness issue, we formulate a joint UAV trajectory and user assignment design problem to maximize the minimum throughput among users. Especially, to strictly fulfill no-fly requirements for certain zones, we treat the trajectory realistically in a continuous manner, which introduces new challenges in handling the highly non-convex problem with the infinite variables and infinite no-fly constraints over the continuous time domain. To facilitate the theoretical analysis on the focused problem, we first adopt the optimal successive-hover-and-fly (SHF) structure to reformulate the problem into a new one with finite variables without loss of optimality which allows tight convex problem approximations at any feasible point with the assistance of our proposed methodology. Then, based on rigorous mathematical analysis on NFZ constraint, problem with infinite NFZ constraints is transformed into solvable one equivalently, enabling an efficient algorithm to obtain a high-quality solution. Finally, numerical results validate necessity of this work on continuous trajectory design.