I think you might be interested in the following important distinction: relativistic mechanics can be built up upon it's own first principles and then derives classical mechanics as a limiting case; and although QM reduces to classical mechanics in a limiting case, QM actual requires classical mechanics for it's very formulation, and thus QM occupies a peculiar place among physical theories. See Landau, volume 3, chapter 1.
Now I will attempt to answer your question, even though in my opinion this question should be asked at math.stack exchange. In the broadest sense, a continuous manifold IS the limit of a discrete set of points, just like the set of real numbers is the limit set of the set of fractions. But after viewing your previous post that you linked to, I would like you to consider the following:
The cardinality of a discrete set of points is the cardinality of the integers, and the cardinality of the set of points on a continuous manifold is at least that of the continuum. There are certainly many consequences of this fact which distinguish the two cases.
Here is another thing you should bear in mind, the FT defines functions by a countable set of coefficients, but FT's only converge up to a set of measure zero.
In QM the FT plays a very important role, and I think the discrete nature of QM is not related to a discrete space time necessarily, but rather to the mathematics of continuous symmetries, i.e. Lie groups. This explains for example the quantum numbers associated with angular momentum.
A Lie group is a group which is also a continuous manifold, and thus we see that already, a discrete set of points can never be a Lie group, because a discrete set of points is by definition not continuous.
Certainly computer methods which make use of discrete approximation have produced good results, but the thing about computer models is you have to be so careful all the time to make sure you are not introducing an error by choosing a bad parametrization for your discrete "mesh".
Let's take a look at the case of fluid mechanics simulations. It is well known, and should be intuitively clear, that for the case of turbulent flows, the density of the mesh at regions of intense turbulence is going to have a strong influence on the results of your computer simulation. There are so many subtle things that are involved with mathematical convergence for these types of situations. It turns out, that there are lots of situations where you just can't simulate the turbulence because the chaotic nature of the turbulence is so sensitive to the initial conditions and the mesh density, that it's very hard to get your simulation to compile (converge) in any meaningful way. I think this example of simulating fluid mechanics really shows the limitations of using discrete methods to approximate solutions of certain problems.