Theories requiring dual continuous and discrete constructs Over the years there have been questions of a similar ilk on MO (e.g., Q1, Q2, Q3) concerning theories in which either continuous constructs or discrete constructs preceded the development of the other or in which one but not both are present. 
I've always been fascinated by quantum mechanics in which notions of the continuous and discrete are necessarily complementary in understanding the math and physics. For example, to characterize the motion of a particle trapped between two walls with a set of discrete, possible energy or momentum states, one has to introduce a continuous probabilty amplitude with the distance between the walls being an integral multiple of the period (or half-period) of the probability wave. The presence of both the discrete and continuous are a necessity to developing a mathematical understanding. The same applies to understanding electrons and the periodic table and the diffraction patterns in the double-slit Young experiments.
A related example is Fourier transform theory in which the discrete Dirac delta and mono-frequency, continuous waves of infinite extent are a necessary duality, not an expediency.
Question:
In what other theories are dual discrete and continuous constructs a necessity?
In response to some comments:
Read Wiki, or better, Feynman's QED for the layman, for a clear explanation of how the iconic double-slit experiment displays the central mystery of QM--the dual wave and particle nature of quantum mechanical objects (wavicles). In a nutshell, a wave-like interference/ diffraction scattering pattern develops for an ensemble of electrons that separately pass through the double-slit over time, each one leaving an isolated  discrete mark on the recording material.
Electrons are fermions whose continuous probability wave amplitudes negate each other (Pauli exclusion principle) when possessing the same state values (unlike bosons. such as photons, who tend to coalesce allowing for the existence of lasers) so that only two electrons with opposite spin can occupy a given orbital of an atom. This accounts for the lack of a classical collapse of the system and the existence of the periodic table.
 A: The Pontryagin dual of the unit circle are the integers.
This implies that the (square integrable) functions on the unit circle are described by a countable sequence of numbers (actually their Fourier coefficients). This is useful e.g. in Harmonic analysis or very practical areas like signal/communication theory. Is it really necessary? I don't know, but understanding how linear filtering of signals work without Fourier series seems quite impossible to me.
A: Decimal (or binary, etc.) representations of lengths along the real line. We intuitively understand the real line as a continuous construct drawn by hand yet to understand the embedded lengths and their relationships, we introduce decimal reps, a series of discrete numbers from zero to nine that represent a resolution of the length into the discrete components $10^{n}$ for the integers $n$, positive and negative--for rational numbers, a finite series or a periodic infinite series, depending on the basis; or for irrational numbers, a non-periodic infinite series. To understand these resolutions, we fall back to the continuous real line.
