I'm looking for info on continuity and discontinuity in maths, and especially on the conditions, definitions, areas of maths etc under which a continuity (e.g. a line) is taken to be strictly equivalent to an infinite amount of discontinuous elements (e.g. points). There is the obvious definition in which a set (?) is continuous if between any two elements there is yet another; but is this everything for all areas of mathematics, or is there any further debate, current or historical?
(By the way, I'm not sure about the terminology. I have little training in maths, would just like some pointers to learn more about how mathematicians think about continuity in various senses.)
A historical example of what you seek was an analysis of continuity by Richard Dedekind. Start with an infinite line, drawn horizontally left-to-right. He noted that, if you choose a point on the line, it partitions the line into two sets of points, such that all of the points in one set lie to the left of all of the points in the other set, and likewise all of the points in the other set lie to the right of all of the points in the first set [actually either of these assertions is provable given the other]; the chosen point may be put into either set. After thinking long and hard, Dedekind concluded that what made the line continuous was exactly the converse of this statement, i.e: IF a line is partitioned into two sets, such that all of the points in one set lie to the left of all of the points in the other set (etc), THEN there is a unique point [which may be in either set] that establishes this partition [i.e it is between the two sets (if you leave it out of both)].
If you need to go beyond continuity of a line, you should be looking into the basics of Topology.