What is a good way to explain the slightly different kinds of continuity?

Question

What is a good way to explain the slightly different kinds of continuity to students? I have in mind these kinds of continuity:

1. A function is continuous at a point. (This also has two sub-kinds: continuity at an interior point and at a left or right endpoint of the domain.)

2. A function is continuous in an interval.

3. A function is continuous. (I.e. it is continuous at every point in its domain.)

My students get these ideas mixed up, no matter how hard I try to distinguish the ideas and the terminology. Do you have a good way to teach these concepts?

Answer 1

In my Calculus classes I really like using the function $f(x) = \frac{1-x^2}{1-x}$ as a running example. First I use it to exemplify how the definition of a function must include information about its domain. For instance, $f$ is not quite the same as the function $1+x$.

Later, I se $f$ to talk about continuity, because it is a good example of a function that is not continuous at exactly one point, even though the limit exists. Then we talk about how you can extend a function by continuity, thus recovering $1+x$.

Finally, we use $f$ to visualize the derivative of $x^2$ at $x=1$. The point I make is that the quotient definition of derivative needs to use a limit, because the quotient itself is not defined. Usually (meaning for quite a few students) the understanding of the continuity issue becomes ingrained as a collateral effect of understanding the definition of derivative. And that kind of is the point in a Calculus class, right?

(As an aside, I also use $f$ in Complex Analysis, now as a function of a complex variable, to explain the subtle notion of a removable singularity. Without a concrete example, they usually miss that there is meat to this definition.)

Answer 2

Those are all the same kind of continuity:

• Given a subset $S \subseteq \mathbb{R}$ and a function $f: S \to \mathbb{R}$, we say $f$ is continuous at a point $x \in S$ iff for each $\varepsilon > 0$, there exists $\delta > 0$ such that, for all $s \in S$ with $0 < \lvert x - s \rvert < \delta$, we have $\lvert f(x) - f(s) \rvert < \epsilon$.
• We say $f$ is continuous iff $f$ is continuous at each point $x \in S$.
• More generally, given a subset $T \subseteq S$, we say $f$ is continuous on $T$ iff $f$ is continuous at each point $x \in T$.

There's no need to give a different definition for "continuous at an endpoint", because it's already encapsulated in the above definitions; any points outside the domain are irrelevant.