In my (non-AP) calculus class last week, I was teaching my students about continuity of a function. Before we started, I asked them what continuity was, and students in both sections started their answer by saying “well, it’s when you draw a function and you don’t have to lift your pencil.” Some spoke of holes and asymptotes. Others spoke of endpoints.
I then proceeded to wow them by saying — all that they said could be encapsulated mathematically. The act of tracing out a function knowing where you’re going to have to lift it can be rewritten with three rules. They weren’t as impressed with that fact as I was, but I still tried to convey “Think about it! You can translate moving your hand across a page smoothly into mathematical statements.”
What’s needed for continuity of a function 
1.
is defined
2.exists
3.
I did the most obvious *you need to memorize this for tomorrow* wink-wink nudge-nudge that I possibly could. I might have even *coughed* the words “pop quiz.”
I just graded the quizzes. Horrible. HORRIBLE.
I got things that show no understanding of the symbols of calculus or what continuity means. Some examples:
(a) function
(b)
(c)exists
(d) one value for
(e) the two-sided limit ofexists
(f) the two-sided limit ofis equal to
(g) the function has to be continuous (you cannot pick up the pencil)
(h)
There are some major notational misunderstandings, but also part and parcel, some conceptual misunderstandings. I mean, for example, “the two-sided limit of
I typed a bunch of these out and we’re going to talk about them in class tomorrow. Hopefully we’ll get to parlay that into a discussion of notation, the precise meaning of math symbols, and the importance of listening to Mr. Shah’s coughs.
Continuous Everywhere but Differentiable Nowhere 
I was recently wondering, with questions like this, if students would recognize their own answer. That is, now that you have a list of all the answers, typed up to avoid handwriting recognition, if you pass it out, how many students would correctly be able to identify which reply was theirs?
Great idea. I’d like to hear how the students respond to this. Do they think you(we) are just being picky or do they understand the differences?
@Nick: I totally wonder if they would. When I put them up on the board, typed, one of them said “oh, that’s mine!” when we got to one. But the rest remained silent.
@Jackie: They didn’t really react or respond. I tried to have them explain what was wrong with each statement, and then I expounded further on that. I definitely don’t think we are being too picky, and I think they don’t understand the differences. The notation and the concept, in this case, go hand in hand.
This reminds me of when my analysis professor asked us to write out the epsilon-delta definition of a limit. Not easy! I still can’t do it without sketching a curve and labeling the little axes regions. That, and your continuity question, is the kind of thing you really have to study to get right.