Past exam of the mathematics course of the University of Cambridge / 2020 / ii / Paper 1 / 12F / c / i / 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 givewhich is impossible. This is the usual diagonal argument behind the noncomputability of the diagonal halting set.