f(n) = { 0, if no Turing machine computes the halting problem
H(n), if there exists a Turing machine, H, that computes the halting problem
}
A function is just any proper mathematical function, a "black box", such that for each input x is given exactly one unique y. A computable function (per Sipser) has some Turing machine M such that M(x) = y everywhere.
It would seem the definitions are also fine for the above contrived example. edit: I'm not sure if n needs to encode a tuple (P, i) for some program P on input i and so forth.
It would seem the definitions are also fine for the above contrived example. edit: I'm not sure if n needs to encode a tuple (P, i) for some program P on input i and so forth.