The semi-characteristic function of $K = \{x \mid \varphi_x(x)\downarrow\}$, that is the set of all indices $x$ such that the $x$-th computable function converges on the input $x$, is the following:
$$ \chi'_K(x) = \begin{cases} 1 & x \in K \\ \text{undefined} & \text{o.w.} \end{cases} $$
This partial function is computable. A colleague of mine claimed that it can be extended to a total function, and the professor, whom had initially claimed this was not the case, agreed, and changed the example.
I can't currently see a way of extending it though, because by substituting any value to the undefined I'd end up with a non-computable function... Is there a way?