computably defined function with non computable range
Your question is a bit unclear, but you're probably looking for this example:
$$
A(x,n) =
\begin{cases}
x &amp; \text{ if program $x$ halts within $n$ steps}, \\
\bot &amp; \text{ otherwise}.
\end{cases}
$$
This is a computable function whose image is the set of halting problems (together with $\bot$).