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.