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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?

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    $\begingroup$ The first one is the definition, the rest are situation dependent. For instance a function can be continuous at only one point. $\endgroup$ – Chris C Nov 2 '14 at 0:25
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    $\begingroup$ I understand all that, but my question is how to better help my students to understand that. $\endgroup$ – Rory Daulton Nov 2 '14 at 0:39
  • $\begingroup$ @RoryDaulton You're lucky that you don't have to explain the difference between absolute continuity, Lipschitz continuity, uniform continuity etc ;) $\endgroup$ – Danu Nov 2 '14 at 12:50
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    $\begingroup$ @Danu: That's not luck, that's sticking to the curriculum ;) $\endgroup$ – Rory Daulton Nov 2 '14 at 12:54
  • $\begingroup$ I have a terminal degree in math, and I am confused by these three "kinds" of continuity as well. $\endgroup$ – Jan Hlavacek Nov 4 '14 at 6:58
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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.)

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    $\begingroup$ That example seems like it could be misleading. The function $f: \mathbb{R} - \{1\} \to \mathbb{R}$ defined by $f(x) = \frac{1 - x^2}{1 - x}$ for each $x \in \mathbb{R} - \{1\}$ is the same as the function $f: \mathbb{R} - \{1\} \to \mathbb{R}$ defined by $f(x) = 1 + x$; the domain is part of the data of the function, not a consequence of the way $1 + x$ happens to be written. Also, that function $f$ is continuous. A removable singularity is (by definition) not in the domain, so it doesn't make sense to talk about whether the function is continuous at that point. $\endgroup$ – Daniel Hast Jun 14 '15 at 14:13
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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.

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  • $\begingroup$ +1 because in my experience it is not clear to all students, that continuity only makes sense for points in the domain... $\endgroup$ – Stephan Kulla Jun 16 '15 at 14:54
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    $\begingroup$ @tampis: My guess is, this is partly due to confusion about what functions are, possibly due to inconsistent or absent definitions. Many high school math textbooks conflate singularities with discontinuities, ask students to "find the domain" of functions (as though it wasn't part of the data of the function), and almost always present functions in the format "f(x) = [some expression involving x]" (leading to confusion about the distinction between the function itself and its values, and mistaken beliefs that functions defined in any other way — e.g., piecewise — aren't functions). $\endgroup$ – Daniel Hast Jun 16 '15 at 15:35

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