What is the difference between 连续， 陆续， 接连， and 一连？ 连续
successive; 
continuous; 
consecutive

陆续 
in turn;
successively;
one after the other

接连
on end;
in a row;
in succession

一连
in a row;
in succession;
running

From what I've read, 连续 is the most widespread term and would be the most accurate word for 'continuous', especially in science, as I have a book called 连续时间金融

From examples, 陆续 is closer to 'one after the other' or consecutively

I can't see what's unique about 接连 and 一连 but they seem different from the other two
As all four words have very similar meanings, here's an attempt to explain the slight differences: 

连续 emphasize on the consecutive-ness - there're no breaks in between.
陆续 emphasize on counting - in a sense of 'keep it coming'.
接连 emphasize on succession / connection.  'one after the other' fits here.      
一连 is the same as 连续 but in a sense of shorter range in terms of usage.