  We analyze the emergence of diffractive focusing in the transition from
discrete to continuous space-time variables. Three types of dynamical equations
are studied in a top-to-bottom approach, starting with the most general system.
First we solve a linear cellular automaton of two species, then a nearest
neighbour tight-binding array and finally the time-dependent Schr\"odinger
equation. All models are shown to produce diffractive solutions for square
packet distributions. The main result of this paper is that in discrete
variables, the nature of the solutions depends strongly on the size of
wavepackets, whereas in continuous variables, diffraction due to
discontinuities exists at every scale. A transition in the number of
participants or cells is identified by means of a measure and the corresponding
phenomenon is further analyzed by a generalization of Wigner functions to
discrete variables and crystals.
