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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)

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    $\begingroup$ The title says "Do you mention that...," but the body talks about "counter-intuitive questions," implying that it's something you'd ask students to figure out. Personally, I think it might be hard to present something like this as a lectur topic without coming off as pedantic, whereas it might be a very nice logic exercise for students to do, e.g., as think-pair-share. What this basically gets at is the nature of quantifiers in logic, which most students need practice thinking about. $\endgroup$
    – user507
    Jul 8, 2020 at 1:40
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    $\begingroup$ AndrewChin, I knew of those ideas when I did mathematical proof and logic. @BenCrowell I'd probably phrase the question to the students as "based on the definition of continuity, what would the empty function be?" Questions related to this one about the empty function would be questions about about functions like Weierstrass, Thomae's or Dirichlet's since those ones focus on the definitions and are very famous counterexamples to intuitions that were found out to be incorrect. $\endgroup$
    – user13544
    Jul 8, 2020 at 2:17
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    $\begingroup$ @Robbie_P_math From your comment above, it seems like this might be what we would call an "Intro to real analysis" course here in the States? Can you add some more context to your question? I think discussing edge cases like "the empty function" is appropriate in such a context, but not in a "calculus class". $\endgroup$
    – Steven Gubkin
    Jul 8, 2020 at 15:36
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    $\begingroup$ You'll want to be a little careful with this, and if for some reason you feel this HAS to be brought up, I recommend saying that (in its domain) the empty function is everywhere continuous and everywhere discontinuous (i.e. don't let "is continuous" be the only vacuous statement you make regarding continuity of the empty function). $\endgroup$
    – Dave L Renfro
    Jul 8, 2020 at 20:57
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    $\begingroup$ @DaveLRenfro Good point: it is strictly increasing and strictly decreasing, it is differentiable on its domain, and nowhere differentiable on its domain, etc. $\endgroup$
    – Steven Gubkin
    Jul 9, 2020 at 12:49

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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.

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    $\begingroup$ (+1) Just getting students to think of functions as something other than input-output devices or algebraic expressions is a major victory in a precalculus or calculus class. Bonus points if students walk away knowing the difference between $f$ and $f(x)$. I'd give another +1 if I could for the second-to-last paragraph, which gives a much more down-to-earth example for calc students. Another couple: does the zero function have a local maximum at $x=47$? and is the function $x \mapsto 1/x$ (or $x\mapsto \sqrt{x}$) continuous? differentiable? $\endgroup$
    – Xander Henderson ♦
    Jul 8, 2020 at 3:38
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    $\begingroup$ In later classes I'd be happy if most students even knew what "constant function" or "identity function" meant when those come up. $\endgroup$
    – Daniel R. Collins
    Jul 9, 2020 at 1:01
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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.

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  • $\begingroup$ My favorite question is the $\log(0)$ one because I can say it's worse than a black hole - isn't a black hole a regular singularity, not an essential singularity? $\endgroup$
    – kcrisman
    Jul 13, 2020 at 13:11
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    $\begingroup$ +1 for "I want to suggest that one should be ready for such a question even in a low-level" And no, I'm not going to worry about whether kcrisman's extended comment is really an appropriate "stack exchange answer" to the OP's question, because I think this advice is important enough to not be buried unnoticed in a comment. $\endgroup$
    – Dave L Renfro
    Jul 13, 2020 at 20:11

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