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Imagine a course barely getting into some topics more theoretical than what is done in the typical very staid first-year calculus course, and the kind of students for whom such a course is appropriate. Here the student might learn a bit about the role of the least-upper-bound axiom for the first time.

In the typical first-year calculus course students are asked to do exercises that say "Why is the following not a counterexample to the mean value theorem?", and are given a function whose domain clearly fails to include one point within the given interval.

A number of theorems of a slightly more "theoretical" account of calculus depend on intervals having no "gaps":

  • The intermediate value theorem;
  • The extreme value theorem (e.g. the function $x\mapsto|x|$ on the set $\mathbb R\smallsetminus\{0\}$);
  • The mean value theorem;
  • The fact that if $f'>0$ on an interval then $f$ is increasing on that interval ($x\mapsto-1/x$ has a positive derivative on its whole domain but its value at $+1$ is less than its value at $-1$);
  • The fact that derivatives have the intermediate value property even if they are not continuous;

etc.

In first-year calculus a "gap" is merely a real number not in the domain of the function, so that, in the exercises mentioned above, the domain is $[a,b)\cup(b,c].$

However, in a theoretical course of the kind referred to in the first paragraph above, as opposed to a first-year calculus course, one wants to define a gap without referring to a point such as the one called $b$ above, which is not in the domain.

Thus: Suppose the domain of a function is the union of two nonempty subsets $A,B$ with the property that every member of $A$ is less than every member of $B.$ For every member of $A.$ Suppose further that for every member of $A$ there is a larger member of $A,$ and for every member of $B$ there is a smaller member of $B.$ That would be a “gap.”

To assert the existence of a gap thus defined is of course logically equivalent, but not pedagogically equivalent (for the kind of students I have in mind) to saying that some non-empty bounded-above set has no smallest upper bound. If you don't see why it is not pedagogically equivalent then I suspect you will entirely miss the point of this question, and maybe you could post a second question asking about that. But to those who do understand the pedagogical non-equivalence I have this question: How does one explain to students in an easy-to-understand way this definition is preferred to one mentioned above involving the point called $b,$ where a "gap" is defined via the existence of an omitted point?

Postscript in response to all (or most of?) the comments: The least-upper-bound axiom and the absence of "gaps" in the sense here defined are plainly equivalent to each other, and both avoid any mention of points that do not belong to the linearly ordered set alleged not to have gaps. The reason I prefer the version involving "gaps" as defined here is that I know that for many students that is easier to understand. That is why I said the two are not pedagogically equivalent. But the essence of the question is: How to explain to students why statements avoiding all mention of points not within the linearly ordered set being considered are better than statements that mention such points?

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    $\begingroup$ I'm familiar with the idea of definitions becoming more abstract for the purposes of generalization. If I'm understanding your question correctly, your abstract definition of a gap addresses the more general case where the domain might be missing a whole interval of numbers, such as $[1,2) \cup (3,4]$ missing the entire interval $[2,3].$ But we could still accommodate this case by tweaking the interval notation definition: $[a,b) \cup (c, d]$ where $b \leq c.$ Is there another case that your abstract definition addresses, that this modified interval notation definition doesn't? $\endgroup$ Commented Jan 31 at 22:25
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    $\begingroup$ @JustinSkycak : That's not what I'm talking about at all. It could be missing just one number or it could be missing a whole interval; both of those are equally valid examples of what we're talking about. But the point is that the definition would not refer to what is missing, but only to what is present in the domain. $\endgroup$ Commented Jan 31 at 22:55
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    $\begingroup$ You assert that your notion should be preferred. In the context of introductory calculus, I don't see why. You are essentially asking that students know the least upper bound property, but this is something which most intro texts tend to avoid, and which isn't really necessary to understand for the vast majority of the material taught in calculus (if you are presenting proofs, it matters for the extreme value theorem---off the top of my head, I can't think of another place where it really matters). What is gained by presenting this more sophisticated notion? $\endgroup$
    – Xander Henderson
    Commented Jan 31 at 23:02
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    $\begingroup$ It's hard to imagine an audience where they will both need this definition and can't handle a definition of path connectedness in the reals. (It is perfectly adequate to say that $a,b \in S\subset \mathbb{R}$ are in the same path component if $[a,b]\subseteq S$. Or a more informal version of the same.) $\endgroup$
    – Adam
    Commented Feb 1 at 16:06
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    $\begingroup$ @MichaelHardy Pardon me for misunderstanding... you spent half of your post providing the context of a first year calculus course, and you only mention any other kind of class in a single sentence, in which it is not clear to me that you mean anything other than a more theoretically oriented calculus class. Perhaps, rather than telling me that I am simply wrong and attempting to make me feel shame for pointing out that confusion, you could simply edit your question to fix that confusion for all? (I will note that I am not the only one to experience that confusion...). $\endgroup$
    – Xander Henderson
    Commented Feb 2 at 15:08

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In my experience, the only way to successfully teach a more sophisticated technique is to present a problem where known simpler techniques fail.

For instance, anyone who's taught algebra to kids will know that if you try to teach $ax+b=c$ equations by demonstrating on a problem that has a single-digit solution, like $2x-1=5,$ then a portion of the students with decent number sense will tune out because they can get by just fine using guess-and-check:

$x=0$ gives $-1,$ too low, go up a bit $x=2$ gives $3,$ tiny bit higher $x=3$ gives $5,$ BAM im so smart i dont even need algebra

The way to avoid this is to present a problem like $9x-13=18$ and say something like "take a moment to see if you can guess the solution to that... yeah, not so easy, huh? Okay, now you see that guess-and-check is not going to work for some of these trickier equations, so let me show you a trick that does work." This way, the kids understand the value in the more sophisticated solution technique that you're about to show them.

It's the same thing at all levels of math. If you want your students to understand why a more sophisticated definition is preferable, then you have to introduce them to a situation where the simpler definition breaks down and the sophisticated definition saves the day.

(And if you can't find such a situation, then I think that's an indication that the definition is not appropriate to use in the class -- though perhaps it could be made appropriate by expanding the syllabus of the class to cover an area where such a situation arises.)

To find such a situation, here's the line of thought I'd start following:

One situation might be when gaps are missing a whole interval of numbers, such as $[1,2) \cup (3,4]$ missing the entire interval $[2,3].$ Your abstract definition accommodates this case, whereas $[a,b) \cup (b,c]$ doesn't.

However, we can easily accommodate this case by tweaking the interval notation definition: $[a,b) \cup (c,d]$ where $b \leq c.$

So, carrying this line of thought further, is there a more nuanced case that your abstract definition accommodates, that we can't accommodate with by tweaking the interval notation definition?


Update: OP has clarified that the key feature of their abstract definition is that it defines a gap without referring to any specific missing points (like $b$ and $c$ in the interval notation definition).

I think a concrete example is still needed to illustrate why it's problematic to refer to such points. Sure, that's a feature of the abstract definition, but it's unclear how that feature "saves the day." (Not saying the feature is useless, just that its utility is not obvious, so an example is needed to illustrate the utility of the feature.)

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    $\begingroup$ @XanderHenderson I think that's for OP to figure out (though I tried to engage OP in discussion about that in the comments). What I'm getting at is that if OP is unable to find such an example, then I don't think it's appropriate to use that definition in the class. $\endgroup$ Commented Feb 1 at 15:10
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    $\begingroup$ You are basically stating that it is up to the original poster to answer their own question. This is a question and answer site---if you don't have an answer that actually addresses the question which is being asked, please don't post a non-answer as an answer. $\endgroup$
    – Xander Henderson
    Commented Feb 1 at 15:30
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    $\begingroup$ @XanderHenderson what on Earth are you talking about? My answer is "find an example where the simple definition breaks down, and if you can't find one, then the definition probably isn't appropriate to introduce in the class." I added an addendum to the answer to make that 100% clear. OP's question was not asking for a specific example -- OP's question didn't even suggest knowledge of need for an example (and OP's response in the comments confirmed that). $\endgroup$ Commented Feb 1 at 15:44
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    $\begingroup$ The notation $[a,b)\cup(c,d]$ where $b\le c$ mentions the points $b$ and $c,$ which are not members of this set. I was attempting to avoid any mention of points not in the set that either does or does not have gaps. The definition of "gap" proposed in my question does that. $\endgroup$ Commented Feb 2 at 15:32
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    $\begingroup$ @MichaelHardy then it sounds like what you need is an example illustrating why it's problematic to mention points not in the set. That's a feature of your definition, but it's unclear how that feature "saves the day". (Not saying that the feature is useless, just that it's use is not obvious, so it's necessary to provide an example illustrating its use.) $\endgroup$ Commented Feb 2 at 15:39
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Your definition of a "gap" is reminiscent of a Dedekind cut. If we further assume $A$ and $B$ are sets of rational numbers and that $A$ and $B$ form a partition of ${\bf Q}$, then we have a Dedekind cut. The cut corresponds to "gap" which in turn corresponds to a new number not in ${\bf Q}$. In this way, we fill all the gaps in ${\bf Q}$ with these newly created irrational numbers.

Your approach seems appropriate to students in a first course in mathematical analysis. To pursue your approach, you could tell students about Dedekind cuts and why the construction was such an important mathematical development.

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  • $\begingroup$ The reason I prefer not to talk about Dedekind cuts is similar to the reason I prefer "gaplessness", in the sense defined here, as preferable to statements about least upper bounds and the like, except that it is even more extreme. $\endgroup$ Commented Feb 4 at 3:43
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    $\begingroup$ @MichaelHardy It is difficult to understand your comment. But the Dedekind cut framework does not invoke least upper bound assumptions. You start with two sets of rational numbers that obey your axioms and which partition ${\bf Q}$. That's all. $\endgroup$
    – user52817
    Commented Feb 5 at 21:24
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How to explain to students why statements avoiding all mention of points not within the linearly ordered set being considered are better than statements that mention such points?

You can't. Because they're not.

To elaborate: You can't make up a new, unconventional definition and expect others to explain why it's better. That burden of motivating it is on you. Aside from classroom math education, perhaps you can have a conversation with a fellow mathematician and see if you can illuminate to them why your definition might be preferred. If that succeeds, then you could cycle back and think about how to repackage it for students.

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    $\begingroup$ The definition stated isn't all that unconventional, and at any rate you seem to misunderstand the main point of the question. Avoiding all mention of points not within the linearly ordered set in question is NOT unconventional. It is done every time the least-upper bound axiom is used. And you state that that is NOT better than methods that do mention points within the set. That part of what I proposed is not new; it is not something I introduced myself. $\endgroup$ Commented Feb 5 at 18:37
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    $\begingroup$ @MichaelHardy: Citation needed. Preferably edited into the OP. $\endgroup$ Commented Feb 5 at 18:39
  • $\begingroup$ Any introductory book on real analysis. E.g. Walter Rudin's Principles of Mathematical Analysis. $\endgroup$ Commented Feb 6 at 16:59
  • $\begingroup$ @MichaelHardy: A citation isn't just the name of a book. It's a short quote, section/page, or in the case of Rudin, a numbered definition. $\endgroup$ Commented Feb 6 at 18:52
  • $\begingroup$ True, but I generally take this to be widely known. I'll see if I can find the most appropriate thing. Possible the chapter titled Three Hard Theorems in Spivak's Calculus can serve. $\endgroup$ Commented Feb 7 at 0:27
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A possibly motivating example might be the following: let $S$ be the set of all positive numbers whose square is not $3$. This description of $S$ seems operationally more straightforward than "the set of all positive numbers not equal to $\sqrt{3}$". To use the latter to decide whether a given number is in $S$, you have to imagine comparing the given number with some number whose decimal expansion most students probably don't remember beyond the first couple of digits and which, in any case, might involve the unpleasant clerical task of comparing long strings of rather unpredictable digits; to use the former, you ask whether the given number squares to $3$ or to a number not equal to $3$, that is, to a number either less than or greater than $3$. While in practice this might also involve a lot of unpleasant work, it's conceptually very clean, and it's easy to visualize what you need to do. It also naturally gives the partition mentioned in your definition.

At the very least this example shows that having to talk about elements not in the set is not always advantageous, and can be disadvantageous.

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  • $\begingroup$ We are converging on the same idea. If $A=\{x\in {\bf Q}: x^2<3\}$ and $B=\{x\in {\bf Q}: x^2>3\}$ the we have a Dedekind cut. The gap so detected is then named $\sqrt3$. $\endgroup$
    – user52817
    Commented Feb 7 at 0:18
  • $\begingroup$ @user52817 Good point! $\endgroup$ Commented Feb 7 at 0:48
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    $\begingroup$ @user52817 Thinking about it more: here we're assuming that there are real numbers, not trying to construct them. So in my answer the partition is into two sets of real numbers, not two sets of rational numbers. Still, there are clearly similarities between what I'm describing and Dedekind cuts. $\endgroup$ Commented Feb 7 at 23:17

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