Mathematics Educators Asked on September 5, 2021

My main question is directly related to the title: "Do you mention that (in its domain) the empty function is everywhere continuous and everywhere discontinuous?" (and a similar question based upon differentiability)

A secondary question I’d ask is, "do you mention the empty function at all?"

My answer: I personally like counter-intuitive questions that really test one’s knowledge of definitions, or at least make one aware of *what* definition they’re using. I might mention it as an aside *as something interesting to note* but I don’t go into it.

(edit notes: updated based upon comment by DaveLRenfro)

The answer by Steven Gubkin is good, but this thought is too long for a comment and is relevant, I think.

I want to suggest that one should *be ready* for such a question even in a low-level precalc or calc class - you really never know what to expect, and I have gotten "naive" questions in such contexts that were actually quite deep, though the students didn't know that! (Then I recruit them to be math majors.)

So although I don't think you should intentionally introduce such topics, you should have a game plan for this. In similar contexts I usually say something like "In this class, that isn't a function, but when you take MATH 12345 we'll really dig into that and it will turn out to be the best function of all, even though it has no values!" The first half is in a normal voice, the second half *sotto voce*. Most students will ignore the wacky math-crazed teacher and not worry about it, but the ones who want to know will come up after class and ask more about this MATH 12345 class you have mentioned.

Lest one think this is concocted, I semi-routinely get questions in my non-major calculus class (which doesn't even have precalculus as a prerequisite in any form) about things like complex numbers, whether vertical asymptotes go to infinity, $log(0)$, and functions defined at just one point. It becomes a running gag in the class that "not in this class! but ..." and actually adds to the bonhomie a bit.

Answered by kcrisman on September 5, 2021

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.

Answered by Steven Gubkin on September 5, 2021

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