I've been wondering whether or not such functions can exist:
(1) $$f: \emptyset \rightarrow \emptyset$$
I can think of only one relation that satisfies this criterion - the empty relation defined on the empty set, namely: $R = \{\}$ but, my question is - is this even a function? The definition of a function says that $f$ is a function iff if two elements have the same first entry, they must have the same second entry - but I think this is not a problem here.
(2) $X$ is non-empty:
$$g: \emptyset \rightarrow X$$
As for this function, there are no possible first entries, and so it is impossible to map anything into set $X$
(3) $X$ is non-empty
$$h: X \rightarrow \emptyset$$
I cannot think of anything that would map something into the empty set. Any suggestions?
Correct; the empty relation on $\varnothing \times \varnothing$ does indeed function defines a function $\varnothing \to \varnothing$.
Incorrect; the empty relation on $\varnothing \times X$ does indeed function defines a function $\varnothing \to X$. Your observation merely shows that the image of this function is empty.
Correct; if $X$ is nonempty, then there are no relations on $X \times \varnothing$ that define a function $X \to \varnothing$.