In "continuous" mathematics there are several important notions such as covering space, fibre bundle, Morse theory, simplicial complex, differential equation, real numbers, real projective plane, etc. that have a "discrete" analog: covering graph, graph bundle, discrete Morse theory, abstract simplicial complex, difference equation, finite field, finite projective plane, etc. I would like to know if there are others. But the real question is: Are there any important "continuous" mathematical concepts without "discrete" analog and vice versa?

8$\begingroup$ I'm not sure what kind of answers you're looking for. Would inner product spaces or normed vector spaces fit the bill? $\endgroup$– Darsh RanjanMar 8 '10 at 21:38

9$\begingroup$ Finite fields are not the discrete analogs of fields. $\endgroup$– lhfMar 8 '10 at 21:39

5$\begingroup$ Maybe this should be tagged as softquestion $\endgroup$– Andrea FerrettiMar 8 '10 at 21:47

5$\begingroup$ Your question is perhaps hopelessly vague. What purpose does the "analogue" have? If it has no purpose, you could call X a discrete analogue of Y for any pair (X,Y). An abstract simplicial complex isn't really "discrete" is it? If it was finite, sure, but if it's infinite, how would it qualify as discrete? $\endgroup$– Ryan BudneyMar 8 '10 at 21:59

6$\begingroup$ I suppose we can also do this in reverse... Are there any important mathematical concepts without continuous analog? $\endgroup$– Gerald EdgarMar 9 '10 at 1:31
A lot of ideas from topology and analysis don't have obvious discrete analogues to me. At least, the obvious discrete analogues are vacuous.
 Compactness.
 Boundedness.
 Limits.
The interior of a set.
I think a better question is which ideas have surprisingly interesting discrete analogues, like cohomology or scissors congruence.

1$\begingroup$ It depends on what you mean by obvious... One example: the interior of a set $X$ of vertices in a graph may very well be defined as that subset of those elements in $X$ all of whose neighbors are in $X$. $\endgroup$ Mar 8 '10 at 22:24

3$\begingroup$ I'd agree that some of these discrete analogues can be vacuous, but isn't that the point? For example, when we study compact sets in topology are we not, at least sometimes, trying to find nontrivial analogues of results that are trivially true of finite sets in the discrete case? $\endgroup$ Mar 8 '10 at 22:41

22$\begingroup$ It amuses me that this answer has been accepted when discrete mathematicians would use analogues of every single one of these. I recommend a look at this post of Terry Tao: terrytao.wordpress.com/2007/05/23/… $\endgroup$– gowersMar 8 '10 at 23:51

3$\begingroup$ I don't know if compactness is a "generalization of finiteness properties" as such, but it certainly gets used as a substitute for finiteness all the time. There are various parts of Banach space/Banach algebra theory where a desire to interchange the order of various iterated limits can be done by judicious appeal to weak compactness of various sets. $\endgroup$ Mar 9 '10 at 8:50

2$\begingroup$ It should also be pointed out that in some sense discreteness and compactness sit at opposite ends of the spectrum of locally compact spaces, so that it's not clear what a "discrete analogue" (as opposed to a "quantitative, finite analogue" might be $\endgroup$ Mar 9 '10 at 8:52
Is there a discrete analogue of the notion of discreteness?

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$\begingroup$ The notion of a discrete metric space being $R$separated for some large $R$ (i.e., any two points are distance at least $R$ apart) might qualify as a discrete analogue of discreteness. $\endgroup$ Jan 5 '15 at 4:42
A timely example would be the lack of a combinatorial Ricci flow in dimensions $n \geq 3$. In principle I think many people believe there should be one, but a combinatorial/discrete formalism has yet to be found.

3$\begingroup$ There is even a discrete analog (Sperner's lemma) to the fixed point theorem. $\endgroup$ Dec 13 '10 at 16:30

5$\begingroup$ I've used the following discrete analogue of the intermediate value theorem in a paper. If you have a function $f$ from the integers to the integers such that $f(x)f(x1)\le 1$ for ever integer $x$, then having $f(x)<0$ and $f(y)>0$ implies there is some integer $z$ with $x<z<y$ or $y<z<x$ such that $f(z)=0$. $\endgroup$ Jun 6 '12 at 12:46
Is "continuous function" an important concept? Does it have a discrete analog?

2$\begingroup$ @Mariano: again, not necessarily. For instance, the (relative) Zariski topology on $\mathbb{F}_q^n$ is discrete, and this is a nonvacuous statement: it has the important consequence that every function $\mathbb{F}_q^n \rightarrow \mathbb{F}$ is a polynomial function. (I think I need a few more rules in order to be comfortable playing this particular game.) $\endgroup$ Mar 8 '10 at 21:51

24$\begingroup$ The discrete analog of "continuous function" is "function". $\endgroup$ Mar 8 '10 at 22:04

3$\begingroup$ Well, there are incredibly interesting discrete analogues of analytic functions (Google should find the notes by Lovász on the subject, for example; this is a whole subject by now) Discreteness of topologies is absolutely irrelevant thereI have no reason to believe the 'canonical' discrete analogue for continuous functions has anything to do with them, either! :) $\endgroup$ Mar 8 '10 at 22:19

5$\begingroup$ I'd say the discrete analogue of a continuous function is one that is continuous in some quantitative way (such as being Lipschitz) on a finite metric space. If the finite metric space is one of a sequence of spaces with unbounded size, this can be very useful. $\endgroup$– gowersMar 8 '10 at 23:53

3$\begingroup$ I did not say that "analytic" is an analogue of "continuous", as far as I can tell. I simply cannot understand what argument there can possibly be supporting a claim of the form 'there is no discrete analogue of X', apart from a standard argument from ignorance. $\endgroup$ Mar 9 '10 at 1:17
It seems to me there is no good (powerful) discrete version of Atiyah–Singer theorem.

$\begingroup$ Maybe GrothendieckRiemannRoch? But honestly I have no idea what exactly it states... At least I know there is no analysis involved in its statement. However, I fear getting something really discrete (= a statement on finite sets) out of it would require some serious constructivization. $\endgroup$ Mar 8 '10 at 22:52
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.
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?
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 NavierStokes 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.
Contrary to the comments appended to the question, I think the notion of analogy can be made precise.
Definition: An analogy of concept A defined in setting SA, is a concept B defined in setting SB such that there exists a generalized setting SX which includes both SA and SB as example settings, and such that there also exists a concept X defined in setting SX which reduces to concept A or concept B when attention is restricted to either setting SA or SB.
In general, an analogy is not unique. A concept could have many analogies, and even for a particular analogous concept there could be more than one way in which it is considered to be analogous.
Example: In Time scale calculus which unifies difference and differential equations, there have been publications with differing answers over how to define the analogy between discrete and continuous transforms. A particular description which encapsulates both the integer and real number transforms may apply to other sets such as the rationals, but a different description might not apply to Q. So an analogy is not just two objects but also the link between them.
Searching Google Web, Google Books and Google Scholar for "no discrete version" OR "no discrete analog" OR "no discrete analogue" OR "no continuous version" OR "no continuous analog" OR "no continuous analogue" produces some examples including a comment that a continuous version of a discrete concept doesn't necessarily enable you to guess the properties of the discrete case.