Are all partial functions partial computable?
The fact that we can encode Turing machines as strings means that there are only countably many Turing machines, so only countably many computable or partial-computable anythings.  Since there are uncountably many partial functions $\mathbb{N}\to\mathbb{N}$ (or even partial functions $\mathbb{N}\to\{0\})$, it follows that, in a very strong sense, "most" partial functions are not partial computable.

For a concrete example, use the fact that, if a set and its complement are both semi-decidable, they are both decidable (exercise: prove this), and the analogous result for partial functions. This means that, if you take any set&nbsp;$S$ that is semi-decidable but not decidable, then $\overline{S}$&nbsp;is not semi-decidable.  So, for example, the function
$$f(x) = \begin{cases}
0 &amp;\text{ if }x=\langle M\rangle \text{ for some TM $M$ that loops on every input}\\
{\perp} &amp;\text{ otherwise}
\end{cases}$$
is not partial computable.