继续 you are doing a task,but you stop in the half way.you need to 继续 finish it.
完成一件事,中途停下来,之后又从这里开始,叫继续
连续 to finish a task in one.
一次过完成的叫连续
持续 keep a long time.like rain,snow....
保持了一段时间,如:下了15天的雨,叫持续15天。
陆续 one by one.i get in the room,then you,then she,then he......
一个一个地