Is the empty function constant Consider this definition of a constant function found in Lee:

Definition. A map $f: X \to Y$ is called a constant map if there is some element $c \in Y$
  such that $f(x) = c$ for every $x \in X$.

If $X = \varnothing$, then $f = \varnothing \subseteq X \times Y$. Taking an arbitrary $c \in Y$, we have $f(x) = c$. Therefore the empty function is constant. Am I right? What is the case if $Y = \emptyset$? This is rather confusing, but I would say the function $f: X \to \varnothing$ is non-constant.
 A: Indeed, the empty graph is (by this definition) a constant map in a vacuous sense, so long as $Y$ is non-empty.  
In particular: if $f:\emptyset \to Y$ is given by $f = \emptyset$, then we can fix an element $c \in Y$ and state for every $x \in \emptyset$, $f(x) = c$.  It so happens that there aren't any elements of $\emptyset$, so the statement stands.
Notably, $Y$ cannot be empty.  In particular, there must exist an element of $Y$ for the definition to apply.
Note also that the only function that maps to the empty set is $f:\emptyset \to \emptyset$ given by $f = \emptyset$.  If $X$ is non-empty, then no relation $f:X \to \emptyset$ can be defined over its entire domain.  That is, $f(x)$ is not defined for any $x \in X$, because there are no pairs $(x,y) \in f$.  So, it's not that $f: X \to \emptyset$ fails to be a constant function, it fails to be a function at all.
A: Yes, the empty function $f \colon \emptyset \to Y$ is constant, for the very reason that you stated. On the other hand, there is no function $f \colon X \to \emptyset$ unless $X= \emptyset$ (and if $X = \emptyset$, then it is constant - see above). The reason that, for $X \neq \emptyset$, there is no function $f \colon X \to \emptyset$ lies in the definition of a function. Since $X \neq \emptyset$, there is some $x \in X$ and - by the definition of a function - there must then be a unique $y \in \emptyset$ such that $(x,y) \in f$. But this is absurd.
