Total function

A total function is one which is defined on all inputs (on all natural numbers). So if a partial recursive function is total, the computation of the value of the function for any input is guaranteed to be successful (i.e., to halt). One problem is that because of the unsolvability of the halting problem, it is impossible to recursively compute/decide which partial recursive functions are indeed total. That means that any proof that a partial recursive function is total must be accomplished using non-constructive techniques (according to most any interpretation of constructivism).