Emergence of the discrete from the continuum An almost eternal theme in Mathematics is the approximation of the Continuum by the Discrete. This core idea goes back at least to Archimedes, and remains active to these very days (and quite likely for the next thousand years) .
You have a continuous structure, say a sphere, and you approximate it as the limit of a series of discrete objects, say polyhedra.
I do not need to provide more examples, I am sure you all have plenty.
But, there is also, though by no means so prominent, a reverse direction.
I am not able to date  precisely when it made its debut in the history of mathematics, but it most certainly appears in the revolutionary work of Fourier: take a discrete function, say the Heaviside step function, and approximate it via series of trigonometric functions:

So, here a discrete object is realized as a limit of continuous ones (in this case smooth functions).
I am especially intrigued by this possibility, which, pushed to the extreme, will depict a mathematical world where the Discrete is an emergent phenomenon, out of a  Continuum
Thus I ask to everybody:

can you list active research in the way of approximating discrete structures via smooth ones?

For instance, a polyhedron via a series of smooth manifolds, or examples in analytic number theory, or patterns in finite combinatorics out of .....(fill the dots)  .
Any thoughtful and possibly well documented answer will get my vote, regardless  the domain chosen (in fact, the more examples I will harvest from heterogeneous disciplines, the happier I shall be).
On the other hand, to get the GREEN the stakes are higher: rather than single examples, a sketch of a general perspective on the Discrete as emerging from the Continuum
ADDENDUM: As pointed out by Andreas Blass, I have implicitly conflated two themes here:

*

*discrete as limit of the continuum

*emergence of the discrete from some background
Point 2) does not seem to necessarily imply point 1) and probably the same applies the other way around. Which one I am interested in? Easy: BOTH OF THEM.
But now that Andreas has already marked this point, the GREEN ANSWER would be, perhaps, a clarification on the relationship between  1 and 2 (inter alia)
 A: Mirco, regarding your second question (emergence from the continuum):
My view would be that the FUNDAMENTAL GROUP of topological objects is an example. It captures discrete properties of a continuous object, e.g. a sphere. Or seen as the deck transformation group, it expresses the discrete structure of the universal cover of a continuous object like torus, sphere, and so on.
A: The emergence of particles as excited states (quanta) of underlying continuous fields in quantum field theory might fit the bill. See The Unquantum Quantum by David Tong, in the section titled Emergent Integers:

Erwin Schrödinger developed an alternative approach to quantum theory
based on the idea of waves in 1925. The equation that he formulated to
describe how these waves evolve contains only continuous
quantities--no integers. Yet when you solve the Schrödinger equation
for a specific system, a little bit of mathematical magic happens.
Take the hydrogen atom: the electron orbits the proton at very
specific distances. These fixed orbits translate into the spectrum of
the atom. The atom is analogous to an organ pipe, which produces a
discrete series of notes even though the air movement is continuous.
At least as far as the atom is concerned, the lesson is clear: God did
not make the integers. He made continuous numbers, and the rest is the
work of the Schrödinger equation.
In other words, integers are not inputs of the theory, as Bohr
thought. They are outputs. The integers are an example of what
physicists call an emergent quantity. In this view, the term "quantum
mechanics" is a misnomer. Deep down, the theory is not quantum. In
systems such as the hydrogen atom, the processes described by the
theory mold discreteness from underlying continuity.
Perhaps more surprisingly, the existence of atoms, or indeed of any
elementary particle, is also not an input of our theories. Physicists
routinely teach that the building blocks of nature are discrete
particles such as the electron or quark. That is a lie. The building
blocks of our theories are not particles but fields: continuous,
fluidlike objects spread throughout space. The electric and magnetic
fields are familiar examples, but there are also an electron field, a
quark field, a Higgs field, and several more. The objects that we call
fundamental particles are not fundamental. Instead they are ripples of
continuous fields.

See Reason for the discreteness arising in quantum mechanics? for more information along these lines. In particular, note that compact operators (or more generally, operators with compact resolvents) have discrete spectra (see here and here). As this answer says:

There are several forms of discreteness in quantum theory. The
simplest one is the discreteness of eigenvalues and the associated
countable eigenstates. Those arise similarly to the discrete standing
waves on a guitar string. The boundary conditions only allow certain
standing waves that nicely fit into the enforced region in space. Even
though the string is a continuous object, its spectrum becomes
discontinuous and is naturally labeled with natural numbers.
Another reason for discreteness comes in with multi-particle systems.
Quantum theory requires that a system that is realized in space-time
contains a unitary representation of the symmetry group of space-time,
the Lorentz group. In fact, you can define a particle in quantum
theory as a subsystem that contains such a group representation. And
because you can't have any non integer fraction of a unitary group
representation, you need to have an integer number of them in your
total system. So the number of particles is also an (expected)
discrete feature, and it plays a role when you talk about single
photons for example, that are either absorbed completely or not at
all.

A: Bill Lawvere has, for a long time now, promoted the idea that the discrete emerges as a limiting case by abstracting from the continuous. At the most general level, this manifests in the distinction between cohesive and constant (i.e. abstract) toposes of sets:
Quantifiers and Sheaves, Continuously Variable Sets, Variable Quantities and Variable Structures in Topoi, Toward the Description in a Smooth Topos of the Dynamically Possible Motions and Deformations of a Continuous Body, Cohesive Toposes and Cantor's 'lauter Einsen', etc...
This line of thought is worked out more precisely with the concept of Axiomatic Cohesion wherein, roughly speaking, a topos $\mathscr{E}$ is exhibited as cohesive relative to a base topos $\mathscr{S}$ via a string of adjoints between them describing how the discrete spaces $X \in \mathscr{S}$ sit inside the larger topos $\mathscr{E}$ of more general (cohesive, combinatorial, etc...) spaces. Lawvere gave some lectures on this topic in Como in 2008 and there are accompanying lecture notes. The nLab page for cohesive toposes is also quite helpful.
To bring things back down to earth a little bit, in Left and Right Adjoint Operations on Spaces and Data Types, he descirbes, in the last section, the following situation.
Suppose we have, in a cartesian closed category $\mathscr{C}$, a commutative ring object $R$ which we regard as the 'one-dimensional continuum'. We can form the correspinding 'complex numbers' ring $C = R[i]$ by defining complex multiplication on $R^2$ in the usual way. Inside of $C$ sits the multiplicative subgroup $S^1$ corresponding to the 'circle'. Finally, since we are in a cartesian closed category, we can extract from the map space $(S^1)^{(S^1)}$ the subspace $Z$ of those endomorphisms of $S^1$ which are group homomorphisms. This $Z$ can therefore be regarded as the 'integers' although it is important to note that it will not necessarily be the usual integers object $N[-1]$ derived from a natural numbers object $N$ in $\mathscr{C}$.
A: (Answer related to the one of Dustin Mixon.)
This appears as relaxation is discrete optimization: if you have an optimization problem $\min f(x)$ with a vector $x$ and the constraint that $x_i\in\{0,1\}$ for all entries $x_i$ of the vector $x$, this often becomes very hard. A common approach is to relax the constraint to $x_i\in [0,1]$. This is a continuous problem which is often easier to solve. There are several instances of the phenomenon of exact relaxation where you can prove that the relaxed problem happens to have a solution which is binary "by accident/magic".
A: In combinatorics, there is no shortage of functions that are hard to compute. Here are two famous examples whose input is a simple graph $G=(V,E)$:

*

*$\alpha(G):=$ size of the largest independent set $I\subseteq V$


*$\operatorname{MaxCut}(G):=$ size of the largest cut-set $C\subseteq E$
Both of these discrete functions are $\mathsf{NP}$-hard to compute. However, there are polynomial time–computable functions $\{\alpha'_k\}_{k=1}^\infty$ and $\{\operatorname{MaxCut}'_k\}_{k=1}^\infty$ that map simple graphs to real numbers (i.e., continuous quantities) in a way that monotonically decreases and converges pointwise to the desired functions.
Warning: While the runtime of the $k$th function is polynomial for a fixed precision (at least under certain models), the exponent increases with $k$. See this thread for more details on runtime considerations for semidefinite programs.
To be explicit, $\alpha'_1$ denotes Schrijver's strengthening of Lovász's semidefinite relaxation of the independence number, $\operatorname{MaxCut}'_1$ denotes the Goemans–Williamson semidefinite relaxation of maximum cut, and the sequences arise from sum-of-squares hierarchies that strengthen these relaxations. The claimed pointwise convergence is a consequence of the identities
$$\alpha'_{\alpha(G)}(G)=\alpha(G),\qquad \operatorname{MaxCut}'_{\lceil |V|/2\rceil}(G)=\operatorname{MaxCut}(G),$$
which follow from results in [Lasserre 2002] and [Fawzi-Saunderson-Parrilo 2016], respectively.
