Space being continuous or discrete

Is there any proof of space being continous or discrete?

A "Turing machine" with finite memory is not, in fact, a Turing machine.
I am aware of this. It is probably better to speak of unbounded memory though. No Turing machine will be able to use an infinite amount of memory (see next answer for why), the infinite memory is only there so that any certain amount of information can theoretically be processed.

Saying a function is "computable" does not mean "it will produce an answer in finite time," it means that it can be expressed as a Turing machine.
No. A Turing machine that does not halt does not really express anything. A computable function can be expressed as a Turing machine which halts when it has the answer.

Right, just like the program to compute the position of every particle in the universe given a time t might not halt, but may still be computable. What is your point?
My point is that it is pointless to speak of a Turing machine with infinite runtime (as explained above), and therefore you cannot define the concept of 'computable numbers' using that (as you seem to do). Therefore, computable numbers are defined as those that can be approximated by a Turing machine to any desired level of precision (or the equivalent digit based definition I gave earlier). In the same vein, describing the universe using a Turing machine with infinite runtime (as would be necessary if it contained variables with a continuous range), is just as silly; If there's no halting Turing machine to calculate the universe, then the universe would not be computable, just as sqrt(2) is not fully computable.

All I said was that the input and output are finite
Why is that? I'd like to see you try to put the values on a continuous number line (say, reals from 0 to 1) each in a finite number of bits.