Solution
= Solution
The function $g$ is not partial computable. It is defined on every input, so if a register machine with code $e$ computed it, then $f_{e,1}=g$ and in particular $f_{e,1}(e)$ would be defined. The first clause in the definition would give
$$
g(e)=f_{e,1}(e)+1=g(e)+1,
$$
which is impossible. This is the usual diagonal argument behind the noncomputability of the <diagonal halting set>.