If the wavefunction is continuous how can the many-worlds be discrete? Preamble for clarity:
The many worlds interpretation is usually used to explain the measurement of a 2 level system ($|0\rangle$ or $|1\rangle$) as:
$$\frac{1}{\sqrt{2}}(|0\rangle+|1\rangle)|\text{device ready}\rangle|\text{env}_a\rangle\to\frac{1}{\sqrt{2}}(|0\rangle|\text{device says 0}\rangle|\text{env}_b\rangle+|1\rangle|\text{device says 1}\rangle|\text{env}_c\rangle)$$
where $|\text{env}_a\rangle$, $|\text{env}_b\rangle$ and $|\text{env}_c\rangle$ are orthogonal (or nearly orthogonal states of the greater environment).
The universe is then said to have essentially split into 2 "worlds", one in which the spin is in state $0$ and the device says it is in state $0$ and the other where the spin is in state $1$ and the device says it is in state $1$.
My question: This picture works for an interaction with a 2 level system but it seems to me that in general one is making an arbitrary discretisation (or coarse-graining of the wavefunction). How does one describe the same process for the measurement of a continuous variable, say of the location of a particle?
Secondary question:
Also, there seems to be an additional difficulty (or maybe its actually the same one in disguise) in that, in reality, we should really say
$$\frac{1}{\sqrt{2}}(|0\rangle+|1\rangle)|\text{device ready}\rangle|\text{env}_a\rangle\to\\\int\int(c^{(0)}_{\theta,\theta'}|0\rangle|\text{device 0}_\theta\rangle|\text{env}_{(b,\theta')}\rangle+c^{(1)}_{\theta,\theta'}|1\rangle|\text{device 1}_\theta\rangle|\text{env}_{(c,\theta')}\rangle)\mathrm{d}\theta\mathrm{d}\theta'$$
where $\theta$ is a variable which we use to enumerate the compatible device and environment states. This illustrates that in reality the device and environment will also become entangled. 
Now the issue seems to be that even some of these states will actually be completely decohered from each other, and may have other observables be incompatible on a macroscopic scale. Hence it seems that we have not 2 "worlds" but 2 sets of "worlds", which it seems may even be continuously connected! (i.e. connected in the sense that for large separation in $\theta$ they correspond to macroscopically distinct worlds but for small separation, they are still coherently connected).
More explicitly I mean that if we project into the measured $0$ "world" in the simplified (standard) example and consider the reduced density matrix for the device we will get
$$\hat{\rho}^{(0)}_{\text{device}} = |\text{device says 0}\rangle\langle \text{device says 0}|$$
i.e. the device is in a pure state of having measured $0$ given that the spin is in state $0$, and so it feels reasonable to call this a unique "world". However, in the more realistic second example, we would find that
$$\hat{\rho}^{(0)}_{\text{device}} =  \int\int c^{(0)}_{\theta,\theta'}|\text{device 0}_\theta\rangle\mathrm{d}\theta\mathrm{d}\theta'\int\int c^{*(0)}_{\phi,\theta'}\langle\text{device 0}_\phi|\mathrm{d}\phi $$
$$\hat{\rho}^{(0)}_{\text{device}} =  \int\int \rho^{(0)}_{\theta,\phi}|\text{device 0}_\theta\rangle\langle\text{device 0}_\phi|\mathrm{d}\theta\mathrm{d}\phi $$
with
$$\rho^{(0)}_{\theta,\phi} = \int\mathrm{d}\alpha c^{(0)}_{\theta,\alpha}c^{*(0)}_{\phi,\alpha}.$$
This is clearly not, in general, a pure state and so the question arises does it correspond to multiple worlds (i.e. are there parts that are totally decohered and behave separately) or is there some way to explain this away and say it is just one? Hence the question arises more generally how does one define a "world"? 
Any explanation to either question would be appreciated.
 A: I think you are asking if, in the Many Worlds view, an interaction can result in a continuum of branches rather than a finite discrete number of branches.  The answer is "yes".  Consider a measurement of a free particle's position or velocity.  Its wavefunction occupies a volume in position-momentum space    (because of the uncertainty principle) with a probability density that varies continuously in both position and momentum.  The outcome of the measurement is any (in MW, read *all) of the possibilities.  So, "branching" is a bit of a misnomer, and is used to make the MW concept easier to visualize.
A: I cannot answer, in general, what the persons believing in the many-world interpretation would say about the problem: the interpretation of scientific theories is not a church with an orthodox position. Rather, I will try to follow the logic behind.
In the example with the 2 level system, the splitting is clearly into two discrete cases, just because we started with a discrete system.
A continuous case would be the measurement of the position of a particle. If we measure the position where a photon hits a CMOS or CCD sensor, we are still discretizing, since we only detect in which pixel the photon hits the surface. The same happens if any digital circuit is used. We see that, in this interpretation, the strange relations between observed object and measurement instrument arise spontaneously from the quantum mechanical theory.
We can imagine a very strange instrument which measures the position of a particle in a really continuous way. For example, representing the speed of a particle with the position of a gauge. In this case, yes, the world splits into a continuous infinity of worlds, very similar to each other.
However, this gives rise to no difficulties. Indeed, as I understood it, the many-world interpretation is based on the idea that the presence of two or more "worlds" (wave packets) cannot see each other, not that they do not interfere. If we take the hypothetical experiments with the gauge measuring the velocity of a particle, we observe a position of the gauge; in order to see any effect of the surrounding "worlds" (wave packets) we should make an interference experiment between the gauges. If such an extremely difficult experiment were done, it would observe the presence of the surrounding "worlds". Indeed, the quantum superposition of states of macroscopic objects has been observed, with extremely difficult experiments. However, when we look at the gauge, we do not make such an experiment, so we do not see the other neighboring worlds: simply because we do not make an experiment to see them.
Personally, I believe that quantum mechanics is an approximated theory emerging from something very strange, still not understood and extremely different from classical physics, even more different than quantum mechanics itself. So, I do not think that there can be anything "true" about any interpretation of quantum mechanics. From my perspective, the "many-world" interpretation is simply the only way to discuss a quantum mechanical problem, without putting additional hypothesis, such as the wave collapse.
