I think that this would be too much of a detour in a regular Calculus class.
You would need to first establish that the empty function is even a function. This requires a really pedantic reading of the definition of a function (a relation from $X$ to $Y$ which is total and single-valued: now you have to really dig in to what those words mean). This is beyond what most calculus students are prepared for: in my experience most of them think of functions as algebraic expressions, and some of them think of them as "function machines". One in a thousand could give a formal definition of a function between two sets. That usually comes up in an intro to proof course, or a discrete mathematics course.
Understanding that the empty function is a function, and understanding that it is continuous and smooth, also uses the concept of "vacuous truth". Namely, for any predicate $P(x)$, the statement $\forall x \in \emptyset, P(x)$ is true. This is also a very subtle logical point which will just sound like mystical gobbledegook until the student has taken a logic/intro-proof course. In such courses, I do encourage my students to be "scholars of the empty set", and think about such questions as whether the empty graph is connected, whether the empty relation is an equivalence relation, etc.
There are plenty of opportunities for the kind of pedantic definition lawyering we want our students to engage in in Calculus. For example, you can given them a formal definition of "increasing function" and ask them whether a constant function is increasing or not.
TLDR: Not appropriate for a Calculus class.