# Terminology, Notation, Ideas

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 $f(x)$ at $x=c$:

1. $f(c)$ is defined
2. $\lim_{x \to c}f(x)$ exists
3. $\lim_{x \to c}f(x)=f(c)$

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 $f(x)$ exists
(b) $f(x)=f(c)$
(c) $\lim_{x \to c}f(c)$ exists
(d) one value for $f(x)$
(e) the two-sided limit of $c$ exists
(f) the two-sided limit of $x$ is equal to $c$
(g) the function has to be continuous (you cannot pick up the pencil)
(h) $\lim_{x \to c}f(x)=c$

There are some major notational misunderstandings, but also part and parcel, some conceptual misunderstandings. I mean, for example, “the two-sided limit of $c$ exists” doesn’t really mean anything useful to us. First of all, it should be the limit of the function, and second of all, it doesn’t say the limit as $x$ approaches something.

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.