There was a talk about students and how they understand the concept of “continuity” of a function. That is a good topic.
One of the examples that was brought up was someone in a graduate program who didn’t understand why the function
“ if is rational and otherwise”
is continuous at . The graduate student said that she couldn’t draw the graph without “lifting the pencil”.
I don’t think that this is a problem with calculus teaching; this person shouldn’t have made it through analysis.
But yes, I agree; sometimes students have trouble with the concept of continuity. So we went on; the idea is that when asked about “what it means for a function to be continuous” students often struggled. Fair enough. The answer that was looked for was: “the function is continuous at if which, of course, means that is defined at to begin with.
Instead, students responded with “keep the pencil on the paper”, “has the same formula (as opposed to a conditional formula)”, “connected graph”, etc.
So I asked “how is the concept of limit defined to begin with” and….see the previous sentence! Such nonsense.
Seriously, if you are going to wave your hands at “limit” (and it may be appropriate to do so) then what is the problem to doing that with continuity?
There is more. Consider the frequently quoted idea that a function can be defined by “formula, text, or a *table of values*”, etc.
We were given something like:
And the first row is considered to be in the domain. The question: “it is reasonable to expect “. You know what the expected answer was, but my question was immediately: “why is it reasonable to expect to take the integers to the integers?”.
Deer in the headlights look.
Then “good point”.
My larger point: a “table of values” only defines a function IF the domain is restricted to the entries in the appropriate row (or column) OR if there is an associated interpolation scheme to go with the table.
Then we moved on.
Example: students were given two examples:
1. Example one: say the temperature at 6 am was 60.0 F and the temperature at noon was 75.0 F. So, was there a time between 6 and noon when the temperature was, say, 68.5 F? Ok, that is reasonable, though students might be confused by digital readouts and maybe a physics student might talk about quantum effects.
2. Example two: The winning team in a basketball game scored 81 points. Does it mean that, at some point in the game, that team had 45 points? Ok, “no” is the correct answer but THIS HAS NOTHING TO DO WITH CONTINUITY, at least as defined by the topology that the calculus students have seen. Example: in a volleyball game (new rally scoring), it takes 25 points to win a game. So the winning team must have had 1, 2, 3, 4,….24 points at one time or another, and that is because in volleyball, scores can only be made in 1 point increments and that is NOT true in basketball.
No wonder students are often confused!
Note: this is not necessarily an attack on the intellect of the person giving a talk. For example, there was a research mathematician at a division I research university who gave the following problem on a calculus exam: , elsewhere. The question: “is differentiable at ? She told TAs to mark the problem “wrong” if the students said yes, because the function changed formula at !!! Note: the question asked “differentiable” and not “smooth”.