The old faithful Calculus text of Stewart has provided me with countless headaches this quarter. I must confess that my calc class is really quite bright. These problems do not occur in your standard freshmen calc class.

The Catch-22:

Stewart presents the definition of continuity at a point. A function is not continuous at “a” if it is not defined at “a.” Got that? Now he goes on to say a function is “continuous” if it is continuous at every point in its domain. This means that the function is continuous. Fine and dandy. Now the bright student pipes up and asks, how can it be continuous since we know it isn’t continuous at 0? Excellent question! A function can be continuous even if it isn’t continuous at every point. Wait! What? You just said that was the definition. No, no. I said a function is continuous if it is continuous at every point *in its domain*.

Etc. Etc. This has been the source of questions nearly everyday since this was covered. In fact, now that we are well into differentiation and things like “a function that is differentiable is continuous” has provided a new source of attempting to get the students to keep all the examples straight of differentiable not continuous, not continuous, not in the domain, is a function continuous at a vertical asymptote, etc.

My fix for the problem is quite simple. Let’s go slightly more general and define continuous on a set. This sounds horrific from the veteran professor’s point of view that can’t even get the students to understand continuous at a point or on its domain, but hear me out. If we start there, then have them do drill exercises such as is f continuous on (1,2) what about [1,2], what about R, what about etc. Then this concept of thinking in terms of sets is not really any more abstract than what they already do. The drill exercises will make sure they understand the concept. Now they can note that “at a point” really means, on the set {a}. Also, we now just specify continuous on its domain, and they realize that this is also a set. I’m not saying we should give them the Dirichlet function and ask whether it is continuous on the rationals or something. But intervals and unions of intervals are things they can do.

Why this fixes the problem: the term “continuous” will no longer be used in an ambiguous sense. Right now the source of confusion seems to come solely from the fact that it hasn’t been drilled into them that secretly you can’t say the word continuous without meaning “on a set.”

The catch-22: As a textbook writer, Stewart most certainly wanted to adopt a convention that would mean he could drop the phrase “on its domain,” (which I concede is explicitly stated, but what freshmen reads their text). It would be rather annoying to both read and write that phrase all the time, especially when it is so natural a convention (what would one even mean by saying continuous and including points that aren’t even in the domain of f?!?). There are also many students struggling with the concrete side of things, so introducing “continuous on a given set” is bound to completely lose those students. We should definitely consider the flip-side, though. This text is meant for a freshmen calc class, and so if drilling this concept and explicitly writing out something that isn’t very clear (to them) is going to help in understanding, then I think it is worth it.

Any fellow calc teachers think this is a good/bad idea? I typed this in a heated frenzy and have been thinking about it for quite some time, so it might not be crystal clear what my complaint is.

October 18, 2008 at 11:32 am

I think your complaint will be clear to anyone who has ever TAed before. I’ve definitely run into similar issues in Calc 3 and Diff Eqs. The more general problem seems to trace back to how we teach math starting in K-12. Every year, we try to make the math slightly more abstract which is more confusing, if you ask me, since every year you tell your students you lied to them before and that what they thought they understood they couldn’t have understood since they didn’t even have the correct concepts. Bam! Suddenly students become math phobic and distrustful and don’t like math anymore because its so pointless to learn since you are going to have to unlearn it later anyway. Okay, this rant had almost nothing to do with your post.

The real comment relating to your post. I think it’s a great idea to define continuous on a set then drill them. This should feel more natural and give them a better feel for what is going on. They should become much more proficient at answering these sorts of “boundary case” questions since there won’t be this gray area of not being sure what’s happening underneath the surface. Honestly, many students get through calc 1-3 and don’t ever even realize that a function is just a (well-defined) mapping from one set to another. They think of a functions as plugging something in for x in a polynomial.

October 18, 2008 at 12:09 pm

There’s something annoying and wrong-headed about the whole approach taken by typical calculus text writers, since “continuous on a set” (meaning continuous at every point of the set according to the textbook writers) doesn’t mean the same thing as “continuous when restricted to that set” (as a topological space in its own right), whereas at a fundamental level it’s really the latter notion which is at all relevant.

For example, the characteristic function of the interval [0, 1] (1 on that set, 0 elsewhere) would not be continuous over the set [0, 1] according to the wrongheaded definition above (a circumspect writer would have to qualify it by inserting a clause about considering only right or left limits at the endpoints). There’s something very confusing and ungainly about such a piecemeal approach, and hell to pay in unlearning later for those who go on to pursue mathematics!

Really, one ought to write a calculus book that gets it right from the mathematician’s perspective. Really, the problem may be that young students have not been properly taught that a function has a domain,

and anything outside that domain is to be considered completely irrelevant for the purposes of considering that function. I think you’re right that for ordinary freshman, it’s enough to consider domains which are finite unions of points and intervals (maybe one can take more liberties with brighter students), and I think it would be pedagogically feasible to give the correct definition for functions having such domains.October 19, 2008 at 12:12 pm

On a slightly relevant side note, I think a major reason that this is allowed to occur is that at major universities, the science and engineering departments have hijacked the freshmen calc courses. They don’t care if their students have any notion of actual mathematics. All they care about is that the student can compute things. Who cares if they know if a function is continuous, differentiable or integrable? As long as they can take the limit, differentiate and integrate.

In effect, the math department is helpless to these demands since 90% of the students in the class come from these departments other departments. E.g. The mean value theorem is not taught here because outside departments think that it is useless, so it has been eliminated from the curriculum.

October 19, 2008 at 8:33 pm

Yeah, that’s definitely the way it was at my school. It also didn’t help that all the professors were “applied mathematicians.” The point is that it doesn’t make sense to teach it as engineering as long as the class has a MA/MATH number on it. I wouldn’t go to an engineering class and expect them to teach me math (in fact the one I took deliberately used all their Excel expertise to avoid math at all costs – such as plugging in random values until you get a min of a function or adding up tons of small intervals instead of integrating). If I take a LIT class, I expect to read books. If I take a math class, there should be math. It shouldn’t matter which students are taking it. They signed up for it; they need to face the consequences now.

My favorite example is my complex professor who knew he was teaching a bunch of physicists and engineers. He told us that on the test any of the 40 trig identities listed in the book may be needed. Even after he showed us how to derive whichever one we needed (using 3-4 basic ones that everyone already knew), people complained endlessly about how we had to memorize all 40. In fact, they went straight to the department head and complained (of course leaving out that fact that they DIDN’T need to memorize them if they would just learn how to make a simple mathematical argument).

The following semester he got fired for putting a proof on a Calc 3 test (derive and show correctness of the curvature formula). Right out of Stewart, page 865 in my book. I was a TA for his class when they removed him of his teaching duties. The students never understood the material better with another teacher (I believe I saw four different Calc 3 classes to compare to), but they hated him for making them do math (i.e. for actually doing some work).

He was the best math professor our school ever had. Ask any math student and they will agree. Too bad 90% of his students were engineers.

October 20, 2008 at 2:48 am

wardjm, may I ask what school is this? And what sort of professor was this: assistant, associate, or full, (or other)? Seems more than a little extreme to fire or let go of someone only for that.

October 20, 2008 at 4:25 pm

I’d rather not name the school for obvious reasons, but essentially it’s true. There were a lot of politics that I was privy to (and much, much more that I wasn’t) that I probably shouldn’t have been, but since I was close to some of the key professors involved — and since I was a TA for the course I heard some inside details. It was very much a matter of engineering students complaining to their advisors and then the department heads throwing their authority behind it and contacting the dean — something very stupid in my opinion since these professors never sat in on the class once! How can they possibly know what’s going on? The dean then went to the math chair who was apparently pressured into removing the professor from teaching the class halfway through. The professor was up for tenure and didn’t get it (apparently since his student evaluations weren’t good enough — go figure). This is pretty much the same thing as being fired. Otherwise, he probably wouldn’t have had his yearly contract renewed. Once things were put into motion by the engineering department no one really stopped to ask questions (such as asking me or any of the math majors about what was really going on — trust me I knew all the math majors at my school and we were all just as confused about what was happening).

Anyway, I drew some conclusions since I continued to TA the course under the new professor (the dept. chair). He removed anything that sounded like a theorem including Fundamental Theorem for Line Integrals, Fubini’s Theorem, Green’s Theorem, Stokes’ Theorem, Divergence Theorem, etc. When they got done with that class, they couldn’t even parametrize a curve anymore. But all the students were so happy because they got amazing grades (after passing tests that they could have passed in high school). Every day they would tell me how great it was to have the other professor gone, and how great this one was, and how easy he made the class for them to understand. When I’d have them solve a problem they would either tell me that they weren’t covering that topic or have no idea how to do it (but that’s ok because it wasn’t going to be on the test). Tell me how easy their engineering classes were after learning no calc 3 (Fluids anyone?)!

October 24, 2008 at 12:31 am

In the calculus course I co-teach, we’ve run across the same problem: We say “continuous” to mean “continuous on its domain,” yet students want to take that to mean the whole real line, for any and all functions.

It sounds like we’ve not had as much trouble with the engineering folks, though. I think all but two or three of my students are engineering majors, who are mostly interested in “how do I use [rule X] or [rule Y]” and not “how can I derive these rules / explain why they work.” Maybe that’s normal for freshmen in general, regardless of major, though.

And to respond to wardjm’s first comment, I’m not convinced students become mathphobic because they think their teachers tell them their previous teachers lied (though it probably doesn’t help). I think (or hope, at least) that they realize that they have to start _somewhere_ and that with each new math class they’re going to learn that the things they learned before weren’t lies per se, but more like specific cases that are easier to wrap their heads around until they’re ready for something in full generality.

On a not-completely-related note, I think mathphobia has more to do with culture. Everytime someone who isn’t a mathematician / scientist / engineer (or training to become one) is even confronted with math that’s more than simple arithmetic, they just throw their hands up in the air. There’s this whole idea that math is somehow for nerds, geeks, and the super smart, and only them.

A high-school teacher once told drove the point home for me by stating that in a conversation with a school superintendent (or some such school official), the official exclaimed how hard math was and how they were never any good at it and never really understood it. The teacher said to me “Why do we live in a society where it’s okay to be be bad at math? No one in education would ever get up and say they were bad at grammar or spelling, so why is it okay to say you’re bad at math?”

October 28, 2008 at 11:28 am

long live continuity on a set.

a related annoyance:

the pre-calc book used here

defines “increasing” and “decreasing”

only on open intervals …

it’s maddening i tell you.