Question

What is the exact difference between total recursive function, primitive recursive function, partial recursive function?

Answer 1

The difference is that partial recursive functions are partial functions. A partial function f:A⇀ B (note the special arrow!) is a function that is, in fact, a function f:S→B where the domain S is a subset of A. For all values in A∖S we have that f is undefined. If S=A we say that the function is a total function. 

a primitive recursive function is a partial function.

Source : https://www.quora.com/What-is-the-difference-between-recursive-functions-and-partial-recursive-functions