co.combinatorics - Are there any important mathematical concepts without discrete analog?

What do You mean by word analogy here? From wikipedia we have ( among others):

The word analogy can also refer to the
relation between the source and the
target themselves, which is often,
though not necessarily, a similarity

So You see similarity in differential equation versus difference equation, but this is mostly matter of aesthetic. In practice if You need discrete equation for continues one, You have to put usually a large amount of work in order to make this analogy working. Of course in principle there is relation among differential and difference equation. But what is important here is not what is similar, but what is a gap between them.

When You say, that discrete case may approximate continues one, in fact You take many assumptions, for example about criteria which constitutes what is that mean approximation.

1. Say what is analogy of holomorphic function? Is discrete complex function on lattice of Gauss integers, good approximation for some complex analytical function? In what meaning? What are criteria? Are all properties of holomorphic function shared by "discrete analogy" and vice versa?




2. For example, it is not true that whole
   theory of differential equations may
   be deduced from difference
   equations. We have
   several equations when we cannot
   find correct approximations, for
   example Navier-Stokes equation has
   no discrete model, at least
   till now. You may say: but chaos is
   analogous to turbulence. Why?
   Because is similar? Why do You may
   say that? Is that someone think two
   things are similar enough to say
   that they are?

Then analogy is so broad in meaning word, that I may say, I can see analogy between every things You may point. It may be very useful as inspiration, sometimes it lead us to great discoveries. For every thing You say is analogous to some continues case, we may have differences between them which allows us to distinguish this cases. They nearly almost are non equivalent even in approximate meaning. They are never the same. It is a matter of criteria, if You may say two things are in analogy.