I assume that by $0$ you mean the empty set ($\varnothing$). I don't know how the book defines a relation (the usual definition is that it's a subset of the Cartesian product of two sets). But unless it mentions that the domain of a relation $R\subset A\times B$ is $A$ then the definition of a function as you present it is different from the standard set theoretic definition of the function and furthermore (iii) is not correct. Under the standard definition of a function (namely that if $F\subset A\times B$ then the $dom(F)=A$) both (ii) and (iii) are correct. To see this you have to carefully examine whether they fall into the definition of a function:
Observe that $\varnothing\subset A\times B$. Also note that for any set $A$, $\varnothing\times A= A\times\varnothing=\varnothing$ and thus its only subset (and possible function) is the empty set. So $\varnothing$ is by definition a relation of $A\times B$ for any $A$ and $B$ and furthermore the only relation if one of the $A$ or $B$ is empty.
Now if $F\subset \varnothing\times B$ then $F=\varnothing$ and there cannot be $(x,y)\in F$, $(x,z)\in F$ and $y\neq z$ (since $F$ is empty). Thus $F$ is a function. Note here that if we assume that for a function $F\subset A\times B$ we have $dom(F)\subset A$, then with a similar argument we show that given arbitrary sets $A$ and $B$ the empty set is always a function between $A$ and $B$ (and thus (iii) is wrong).
Now for (iii) under the standard definition, if $A$ is not empty then $F$ has to be non-empty since its domain is not empty, but on the other hand $F=\varnothing$ (as a subset of the empty set). Thus $F=A=\varnothing$.