Yes, I mean "total recursive", I'll edit the question, thanks!

The problems you mention are not recursive, the question was about total and not PR functions. Do you know any?

I was thinking in the definition of the function, A(0,n)=n+1, etc.

1) Isn't PR decidable very weak? It can't even answer if $f(0)=0$, where $f$ is the input function. 2) What is the relationship between the existence of discontinuous predicates and Rice-like PR theorems?

Very interesting! 2 questions:

Given a TM M with index e define a PR function f(n) that outputs 1 if the computation M(0) halts in exactly n steps, and outputs 0 otherwise. The property P(f) holds iff the computation M(0) does not halt.

This is an interesting answer. In an eventual "Rice Theorem for PR functions" we have to clarify what a "quantified, nontrivial question" is. For instance, the following "$\exists n : f(0)=1+n$" is not such question because it is quantified and nontrivial, but decidable.