Inherently discrete concepts Are there any concepts which are naturally defined only for the integers and so far has resisted any attempts at extension to other fields such as rationals or reals?
Does not meet criteria:
Cardinalities of sets
n! / Gamma function
Differentiation / Fractional differentation
 A: I've not yet seen, that the "rank of a matrix" has been interpolated to fractional ranks (but I'm surely not very profound with literature...)
(side remark: also your question focuses on whether it has "resisted to attempts" to interpolate ... such attempts may or may not exist, but to know this needed even a bigger radius of insight into literature and non-literature manuscripts...)
A: Turing machine is a discrete model.
A: The Arithmetic derivative.
A: I don't know if this works, because someone may actually have done something along these lines.  I'm by no means an expert on mathematics in general.  That said, some of the concepts of number theory may well work.  For example, what would the concept of a prime number, composite number, "versatile" number (highly composite number), etc. mean in terms of real numbers (or rational numbers for that matter)?  5 has only 2 positive integer factors (other than 1), but it has an infinity of factors in just the rational numbers.  So, if such an extension of the concept of prime number or composite number exists, I don't know how it works at least, and it would take some explaining.
The Fundamental Theorem(s) of Arithmetic don't seem extendable to the reals, since we don't seem to have the notion of a prime number in the reals, in the sense that some numbers exist in the real numbers which have exactly two factors.
I'll add that I know of at least two concepts which can get defined for the integers, but simply can't get extended to the rationals or reals (and never will legitimately).
For any given number n, there exists a least number o, such that o>n in the integers, where ">" indicates the usual ordering relation of "greater than".  There does not exist any such number in the rationals or reals.  One might say that in the integers, every integer has a distinct-least-upper-bound or distinct-supremum.
For any given number m, there exists a greatest number l, such that m>l in the integers.  There does not exist any such number in the rationals or reals.   One might say that in the integers, every integer has a distinct-greatest-lower-bound or distinct-infimum.
In other words, given a "direction" which either takes towards larger numbers, or smaller numbers, for any number x, there exists a "next" number "y" and a previous number "v".  This does not hold true for the reals or rationals.
A: The principle of mathematical induction.
Reduction modulo $n$, as a map of rings.  It exists only as a map of additive groups for $\Bbb{Q}$ and $\Bbb{R}$.
