Slick way to define p.c. $f$ so that $f(e) \in W_{e}$ Is there a slick way to define a partial computable function $f$ so that $f(e) \in W_{e}$ whenever $W_{e} \neq \emptyset$? (Here $W_{e}$ denotes the $e^{\text{th}}$ c.e. set.) My only solution is to start by defining $g(e) = \mu s [W_{e,s} \neq \emptyset]$, where $W_{e,s}$ denotes the $s^{\text{th}}$ finite approximation to $W_{e}$, and then set
$$
f(e) = \begin{cases}
\mu y [y \in W_{e, g(e)}] &\text{if } W_{e} \neq \emptyset \\
\uparrow &\text{otherwise},
\end{cases}
$$
but this is ugly (and hence not slick).

Perhaps the reason your solution seems ugly to you is that you appear to be excessively concerned with the formalism of representing your computable function in terms of the $\mu$ operator. The essence of computability, however,  does not lie with this formalism, but rather with the idea of a computable procedure. It is much easier and more enlightening to see that a function is computable simply by describing an algorithm that computes it, and such kind of arguments are pervasive in computability theory. (One can view them philosophically as instances of the Church-Turing thesis.)
The set $W_e$ consists of the numbers that are eventually accepted by program $e$. These  are the computabley enumerable sets, in the sense that there is a uniform computable procedure to enumerate their elements. 
We may now  define the desired function $f$ by the following computable procedure: on input $e$, start enumerating $W_e$. When the first element appears, call it $f(e)$. 
It is now clear both that $f$ is computable and that  $f(e)\in W_e$ whenever $W_e$ is not empty, as desired.
