Define $\Phi_e^K(x)$ to be the output of the eth Turing machine that has K (the diagonal language) on its oracle tape and x on its input tape.
Is the map f: (x,e) $\mapsto$ $\Phi^K_e(x)$ a computable function?
let $\omega$ be the naturals.
recall a f: $\omega \times \omega$ $\rightarrow \omega$ is a computable function if there exists a turing machine that given input in $\omega \times \omega$ always halts and upon halting, outputs an integer in $\omega$ on its tape.
I reason that f is not a computable function since the machine that exists must carry K in its description. However K is infinite.
even if K were replaced by empty, we would still have an uncomputable function since there exists e s.t. there exists x s.t. f(x,e) never halts. since there are such turing machines (e.g. the machine that recognizes the halting problem) right?