I am not completely sure that why computable functions is defined as a subset of all functions from/to Natural Numbers, but as far as I know, computation comes from real (physical) world so instructions, inputs and outputs should be finite (intuitively, because a human or machine can not complete a task which contains infinite amount of steps, with finite steps (or finite amount of time). I wonder that can I imagine a computing model which can accept real numbers as its inputs and outputs, and a program is again a real number (or equivalently all programs in this computational model is set theoretically isomorphic to real numbers).
For example, I imagine that a one dimensional empty space (physically) and a particle in that space which moves with a constant velocity. And I see that universe as a computing machine (it takes a real number which is bigger or equal than $0$ representing the time (I think that nonnegativeness is not a problem for this situation because again nonnegative real numbers set theoretically isomorphic to real number) and it gives anothor real number which represents position of the particle). Of course this universe computes only constant slope functions ($ax+b$). This example can be extended to much bigger subsets of the all functions from $\mathbb R$ to $\mathbb R$ (I guess that it should be a proper subset because of the cardinality issues). Although, this model is an abstract model (because of the measurement issues) theoretically it looks like that it is possible.
Lets call this computational model as $\mathbb R$-computable. How can I determine which functions is $\mathbb R$-computable?