 Solution

Fix $x_0\in X$. The property defining ${\mathbf{Inf}}_X$ depends only on the partial function computed, so it is an index set. It is nontrivial: an index of the nowhere-defined function is outside it, whereas an index of the total constant function $n\mapsto x_0$ has infinite domain and range $\{x_0\}\subseteq X$, and hence lies in it. The Rice theorem now shows that this infinite-domain range-restriction index set is not computable.