“Continuous mathematics is, roughly speaking, math based on the continuous number line, or the real numbers. The defining quality of it is that given any two numbers, you can always find another number between them – in fact, you can always find an infinite set of numbers between them. Building up from numbers, a function in continuous math can take the form of a perfectly smooth curve without any gaps or breaks. In theory, you can talk about a function as a set of pairs – f={(x,y) | y=f(x)} – but you can’t even show an exhaustive list of the pairs that make up the function, even over a finite section of the function.>

In discrete mathematics, you’re working with distinct values – given any two points in discrete math, there aren’t an infinite number of points between them. If you have a finite set of objects, you can describe the function as a list of ordered pairs, and present a complete list of those pairs.”

http://scienceblogs.com/goodmath/2007/03/basics_discrete_vs_continuous_1.php