Nonempty set mapped to $\emptyset$ and vice-versa

A function from a set $A$ to a set $B$ is a subset of $A\times B$ satisfying certain conditions, one of which is that its domain is $A$. If either $A$ or $B$ is empty, $A\times B=\varnothing$, and $\varnothing$ is therefore the only subset of $A\times B$. If $A\ne\varnothing$, $\varnothing$ is not a function with domain $A$, so you’re quite right about $(a)$: there are no such functions. If $A=\varnothing$, though, it’s a different story. The domain of the function $\varnothing$ is $\{a:\langle a,b\rangle\in\varnothing\}$, which is ... ?

HINT: Recall that a function from $A$ to $B$ is a subset of $A\times B$, whose domain equals $A$.