There's already a ton of useful answers here, but I figured I throw in another stimulus for reflection.
From the way you ask your question (which is very legitimate), it looks to me as you consider numerical (discrete) solutions as if they had nothing to do with the real (continuous) problem. Or, in other words, that there's no need of real numbers and symbolic calculus if all you will do is use numerical approximations on a computer. Well, this could make sense once the methods are already available and safe to use, but that would limit the sets of problem that we can (numerically) solve to those that we already know how to solve. Let me explain.
What is a numerical method? In few words, it is a procedure that, under certain conditions, provides us with an approximate solution to the problem at hand. But what does it mean that the solution is "approximate". Well, sounds pretty clear: it is "close" to the true solution of the true problem. But how can you measure the distance between the approximate (discrete) problem and the real (perhaps continuous) problem if you don't position yourself in the setting of the real/continuous model?
Numerical methods, in order to be safe to use, must come with a "certificate" (a proof, in math language) that guarantees that, if some assumption are satisfied, then the method will work. This is crucial, otherwise we're just pressing a button and hoping that the number that the computer spits out is actually trustworthy. But again, in order to produce this certificate, you must be able to compare the discrete and the real problem. Since the discrete problem is formulated in a setting that is (usually) a strict subset of the setting of the continuous problem, your only hope is to compare the two problems in the continuous setting. Therefore, you need to be able to work (at least symbolically) in the continuous world, which is a superset of the discrete one.
Once you know (i.e., prove) that the procedure that you came up with makes sense in the continuous world, only then you can temporarily forget about the continuous space structure and only work in the discrete one. But even then, philologically, an approximation can be called as such only if there is something else to which the approximation is related (and hopefully close). So, even though you don't need to be able to manipulate the object in the continuous setting, you still need to be able to formulate the problem in the continuous setting before you start talking about the approximation.
Hope this made some sort of sense.