Counterintuitive consequences of standard definitions

Question

Let me motivate my question with the following situation. While teaching the concept of continuity, I usually start with motivating the concept. Then, when we see that there is an important and intuitive concept here, we start formalizing it. After arriving at the standard definition, it is usual that some counterintuitive facts surface. For example, with the definition I use for continuity, the following facts cause problems:

• A function defined at an isolated point is continuous at that point (trivial "empty" consequence of the $\varepsilon$-$\delta$ formulation)
• The function $f(x)=\frac{1}{x}$ is a continuous function (it is continuous at every point in its domain). (This last one is also a problem because some calculus books make ambiguous statements about its continuity.)

Students usually find these statements strange, and I try to convince them that the definition is so good that we should accept these facts. Making a new definition which excludes these facts does no t make sense. I am, however, not too satisfied with this explanation.

• Can I do better?
• How do you handle this situation?
• What do you do in a similar analogous situation?

This question is (in a distance) related to this other question on empty cases in definitions.

Answer 1

It is unfortunate that the students are taught that:

a continuous function is one for which you can draw the graph with a single stroke of the pencil.

Of course, this is not consistent with the general definition of continuous as it intersects first semester calculus. We say:

a function is continuous if it is continuous on its domain. Moreover, this means $f(x) \rightarrow f(a)$ as $x \rightarrow a^{\pm}$. We require only left or right limits as appropriate to boundary points of the domain. If the point is isolated then the function is continuous at such a point by definition.

As the OP points out $f(x)=1/x$ is continuous, but students are taught this is discontinuous in some algebra or precalculus courses. Of course, you do need to lift your pencil to graph $y=1/x$. So, how can we reconcile this? Frankly, we can't. The previous course was in error. That said, I try to smooth things out by emphasizing that when we restrict the focus to connected domains aka intervals then their intuition is restored. It is true that:

the graph of a continuous function with a connected domain is drawn without lifting your pencil.

The old way (precalculus) of thinking is that by default a function has domain $\mathbb{R}$ then as we discover defects in the formula $f(x)$ points are removed due to various types of discontinuities; removable, jump, vertical asymptotes are the usual suspects. This is intuitive, but, this is not how we carefully work with functions $f:A \rightarrow B$. The points where $f$ is not defined are not taken away, there not even there at the outset. I face this issue with both eyes open and don't shy away from it in calculus because I want students to pay more mind to the idea that a function is not just a formula. The domain and codomain must be specified to be careful. Our theorems hold for sets of numbers, the type of set matters. Connected is key.

Answer 2

Here's an analagous situation: http://en.wikipedia.org/wiki/Clopen_set. Intuitively, open and closed seem like mutually exclusive concepts, but their definitions allow for 'clopen' sets.

More generally, I'd argue that it's both useful and more elegant to have definitions which are inclusive, rather than exclusive. In other words, if we tried to define continuity in such a way that we removed all the seemingly "weird" (i.e. unusual, vacuous) cases, then we'd end up with a convoluted definition. And that's not to mention the difficulty of systematically identifying and eliminating the "weird" cases.

Answer 3

I don't have idea to do better, but some to explain such consequences of definitions.

Some thoughts:

• Definition have, of course, a historical development and come from a lot of examples and attempts to prove theorems and properties of examples. If students are not happy with the definitions, you can give them note how mathematics works and that definitions are coming at last in order to summarize technical properties needed for some theorems.

• Mathematicians should have intuition to work. However, intuition should come from working a lot with definitions and experience; "real world ideas" can sometimes be very confusing. If everything comes from real world intuition, why do you need people doing mathematics?

• An important point is: What are definitions good for? Of course, you can introduce some definition and find out what functions hold that property and which don't. But if you define something, you want to work with it! The task is not to find out which functions are continuous, but to find out: For which function can I use the 100 theorems which holds if the function is continuous! (A similar example: When I was in school, I always wondered if zero is a natural number of not and I believed that mathematicians are doing research about that topic; but later I realized that this is insignificant: There are theorems stating: Something holds for every natural number and sometimes, I have to exclude zero and sometimes zero has to be included.)

• Sometimes, there are (like in your first example) "empty" consequences from a definition. Good point is that it does not contradict the definition. At first, to say that these function is continuous is like an answer to a question no one would like ask. At that point, you can explain (like I said before): You are interested to use some often used arguments like: If $f(x)\neq 0$ holds for some $x$, then this holds in a sufficiently small neighborhood of $x$. This is true if the function is continuous. If you have a function which is continuous only as an "empty" consequence, even the question is useless, since there is not neighborhood of $x$, where the function is defined. At that point, students might suggest a modification of the definition of continuity to only define it for domains having not singular points. You can then say that is is possible to include that, but then the definition is longer and for all excluded cases, the consequences are not wrong, but irrelevant. And you can give an outlook that there a generalizations (in topology) where continuity can in general not be related to some real world intuition and you don't want to get rid of the additional empty examples (e.g., $f(K)$ is compact if $K$ is compact and $f$ is continuous). [I'm not sure how much one wants to go into detail there.]

• According to your second example, I must say that I was very annoyed as a student for such "counterexamples", since most students would say: Okay, let's look at the same function on a different domain and everything works fine. Plus, you normally have the definition of continuous in one point and continuous as a function on a given domain.

Answer 4

You can do better using uniform continuity: $f$ is uniformly continuous on $[a,b]$ if $$\forall\epsilon\ \ \exists\delta\ \ \forall x,y\in[a,b], \ \ |x-y|<\delta \rightarrow |f(x)-f(y)|<\epsilon$$

This has several advantages:

1) It makes more sense in applications. E.g.: If astronomers want to determine the volume of a star to within $\epsilon$, how accurately do they need to determine the radius? It does not help to give a $\delta$ which depends on the exact and unknown $r$. It makes more sense to give a $\delta$ which works for any $r$ within the preliminary estimate of $[a,b]$.

2) It allows a more honest proof that continuous functions are integrable. The proof from pointwise continuity appeals to uniform continuity anyway, either leaving that implication unproved or providing a proof too advanced for calculus students to appreciate. Starting from uniform continuity instead provides all the data for an integration algorithm, and the proof follows.

3) It avoids your counterintuitive statement by leading to the sensible statement "for every $a>0$, the function $1/x$ is uniformly continuous on the intervals $(-\infty,a]$ and $[a,+\infty)$".

This (especially #2) is Errett Bishop's insight.

Answer 5

In the end, definitions are for use. If in some outlandish situations, which aren't "normal use" of the definition, something strange happens, so be it. It's not worth "fixing" (i.e., complicating) the definition for such cases.

The continuity of $\frac{1}{x}$ is fine, that the function goes to infinity at $x = 0$ is of no consecuence as long as you stay away from that point. A function defined just at one point is of no real interest when using continuity, so it is also fine.

Answer 6

I'll attempt to answer your question in reverse order.

What would I do in a similar situation?

The choices for what we now consider standard definitions were refined countless times before the a single definition was accepted. It is easy to forget that and it is partly our job as educators to explain this from time to time to the students (least they start thinking mathematics is a static discipline). So, such situations were students show a distrust in a particular definition that we know is a really good one is an excellent opportunity to engage the students with the dynamic nature of mathematics. I usually encourage the students then to scrutinize the definition, making their objections explicit. Depending on how much time I wish to spend on such things I then decide whether or not to look into their objections more carefully or not.

How do I handle this situation?

The two particular difficulties you mention I would treat as follows. Difficulties on whether or not $f(x)=\frac{1}{x}$ is continuous are the result of being careless with specifying the domain. This is thus an excellent opportunity to explain that the question "is $f(x)=\frac{1}{x}$ is continuous" is meaningless until one specifies the domain. If there are some books that fail to do this properly, then it's a good opportunity to warn the students of lousy books.

The second situation, that of vacuous situations, can indeed be quite confusing but I would not spend too much time on it since such things rarely show up in nature. Moreover, once students are a bit more proficient with mathematical arguments they will easy dispense with such trivialities themselves, so the problem solves itself. If students are really troubled by this I would tell them to spend 15 minutes thinking about it and resolve it themselves.

Can I do better?

Well, if one can no longer improve, then one is dead, so yes, you can also do better :)

The particular notion of continuity is actually a very tricky one, at least historically. Our intuition for what continuity means is quite bad. It is common to say that a function is continuous if its graph can be drawn without lifting the pen from the paper. This is wrong of course, as it more accurately describes functions that are infinitely differentiable at all but finitely many points. It is actually quite difficult to geometrically understand continuous functions. This is something the students should be able to relate to - after all it is difficult for them to understand this concept. Also, historically, mathematicians struggled long and hard with this concepts with many disagreements between Lagrange, Cauchy, Bolazano, and Weierstrass. Some of these historical accounts can also be conveyed to the students. It can also be mentioned that the standard Cauchy definition is not the only way to formalize the notion of continuity, just to tell the students that more than one road exists.

Answer 7

When I first learned about $p$-adic metric which leads to a situation where every triangle is isosceles, two open balls can intersect only if one of them is contained in the other, I was shocked. I was told metric spaces are not easy it can have such examples. Later I realised that the definition of a distance function the three requirements for a function to be called a metric are not restrictive enough to exclude such counter-intuitive $p$-adic metrics.

Now it is a matter of convention whether Archimedean condition is to be imposed for all metrics. If one can develop a theory without that extra assumption and even if the counter-intuitive $p$-adic spaces other wise satisfy many other properties we can live with that.

Same argument can be given for the requirements on a topology on a space: should Hausdorff separation condition be included in the definition? If the theory is rich enough to provide lot of examples where many properties can be proved then there is no need to bring in that condition. Let a few counter-intuitive cases co-exist peacefully.

Answer 8

The case of continous functions is a little bit different than the case of natural numbers.

The basis is the very local definition of continuity at a point, which shouldn't raise any discussions. Then there are two ways to derive global definitions which serve different purposes:

• define continuity for all points of a given interval (or other set). This serves the purpose to inspect that very interval (or set). Why? Don't know, ask the physicist who wanted to know.
• define continuity for all points of the functions domain. This serves the purpose to understand the function as a whole better.

You could also define continuity for all points in the convex hull of the function's domain. You can ask the students, what purpose this definition serves. Answer: It determines, if you can draw the graph in one stroke (possibly taking infinitely long or being infinitely fast).

What definition you take, depends on the purpose. It's irrelevant, if some of its examples are exotic. It's relevant, if some of its examples should be counterexamples.