TITLE: What is the difference between discrete and continuous mathematics?
QUESTION [13 upvotes]: I am studying computer science and this has me absolutely flummoxed. The definition I can find is that discrete data is countable and that continuous is uncountable.
Examples are given stating that integers are discrete, and real numbers are continuous. Rationals it states are controversial.
I understand topics that fall within the category of discrete mathematics, I don't get why they do. Especially set theory, what if my universal set is the real numbers? How can I have a continuous realm of values and consider my structure to be discrete?
Considering both integers and reals belong to an unlimited realm of possibilities, I fail to see how either one is considered countable; they are both infinite. How would you define discrete mathematics and how would someone recognize discrete or continuous mathematics when faced with it?
Note: My knowledge of mathematics is discrete and very limited so please explain this as you would to a 6 year old.

REPLY [3 votes]: To address two of the questions raised:
1.  What is a discrete set and how is this related to discrete mathematics?
Discrete sets consist entirely of isolated points, i.e. sets for which one can find a small enough neighborhood so that only one point of the set is contained in it -- the intuition is that the points are spread apart with a minimum distance between points.
Discrete mathematics is that which is done using finite methods typically using just the integers (e.g. combinatorics, elementary number theory) or at most a finite subset of the rationals, (e.g. discrete probability theory).
2. Are the rationals a discrete set?
No, the rationals are not a discrete set even though they are countably infinite.  This is because the rationals have no isolated points---you can always find a nearby rational number as close as you like.  This is called the Archimedean property of the rationals, and you can see it by asking for any tiny fraction, say 1/100,000, can I find a smaller one?  Sure: 1/100,000,000.  So there is no neighborhood that you can put around a rational point that you could not find another rational within.
However, any finite collection of rational numbers is discrete since when you only have finitely many rationals, then there is a minimum distance between any two rationals in the set, so one can find an interval smaller than this which will guarantee only one rational is in it at any given time.  This means a finite set of rationals is an isolated set, and therefore discrete.