Can a well defined function in Intuitionism come only from the natural numbers? I've seen that there is a problem in the definition of the functions if the domain is the reals because there are real numbers that does not have a value. So does it make sense to say that the only way for a function to be really defined in Intuitionism is if it's domain is the natural numbers?

There is a vast literature on intuitionistic and constructive approaches to the foundations of analysis. These approaches start by viewing a constructive real number $x$ as a process that produces (hopefully increasingly better) approximations to $x$. Bishop's work on constructive analysis and work by Weihrauch and others on computable analysis are good places to start in the literature on this subject.