TITLE: Does the map $f:\emptyset\longrightarrow \{0\}$ exist?
QUESTION [5 upvotes]: What function $f$ maps in the following way:
$$f:\emptyset\longrightarrow \{0\}$$

REPLY [17 votes]: There is exactly one such function, which is the empty set.
Explanation:
By definition, functions $f : A \to B$ are certain subsets of the cartesian product $A\times B$ (for the defining property, see below). You can imagine this definition as identiying $f$ with its graph.
In your case, $A = \emptyset$, $B = X$ and $A \times B = \emptyset \times X = \emptyset$, whose only subset is the empty set.
The defining property is that for every $a\in A$ there is a unique element of $f$ having $a$ as its first component. In our case $f = \emptyset$, this condition is vacuously true.