Let $f_1 $ and $f_2$ be two functions from $\mathbb{N} \rightarrow \mathbb{N}$.
$f_1$ is turing-computable while $f_2$ is not.
$h(i)$ is a function that returns $1$ when $f_1(i) = f_2(i)$ and $0$ else.
The challenge is to construct $f_1 $ and $f_2$ so that $h(i)$ is computable and additionally so that it is not.
My thought process is that for either scenario we have to use the computability and non-computability property and pass that forward to $h(i)$ but I am not sure how.
One thought I had to make $h(i)$ uncomputable is to define $f_2$ and $f_1$ on a for a small set of numbers and let $f_2$ be uncomputable on all other numbers. This way I cannot compute if $f_2 = f_1$ so I cannot compute wether $h(i)$ outputs $1$ or $0$.
But I dont know how to make $h(i)$ computable given the requirements that $f_2$ be uncomputable.
Can anyone nudge me in the right direction ?