In order to teach continuity of real valued functions $f:D\to\mathbb R$ one may start with the (in some sense wrong) intuition

$f$ is continuous when its graph can be drawn without lifting the pen.

I guess this is the intuition many students start with when they first encounter continuity. However, this isn't the formal definition of continuity.

Thus, my question is: How can we motivate the formal definition of continuity starting from the above intuition?

To clarify my question: Imagine yourself in a time when there was no formal definition of continuity (when for example people thought that functions fulfilling the intermediate value property are continuous). How would you argue that the formal definition of continuity must be of the form we have it today?

Thereby, with "formal definition" I mean either the definition in terms of limits of sequences or the epsilon-delta definition.


My attempt for answering the question: From the above intuition one may deduce several properties, continuous function should have:

  1. Continuous functions should fulfill the intermediate value property.
  2. The graph of a continuous function should be a connected subset of $\mathbb R^2$. (Although this is not true for continuous functions in general, this property makes sense when one starts from the above intuition)
  3. Small changes in the input result in small changes in the output.
  4. ...

Now we take this property as the definition of continuity which is (in our opinion) the most characteristic property of continuity. Here we might chose property (3) from which the $\epsilon$-$\delta$-definition can be deduced (note that (3) is similar to the definition of continuity by Cauchy which is one of the first formal definitions of continuity).

What do you think about this kind of explanation? When it is in your point of view a good explanation: Why should one use property (3) as the most characteristic property for continuity and not (1) or (2) or another property?

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    I am reminded of part 2 in Thurston's On Proof and Progress in Mathematics; the thirty seventh definition (start of page four) is both humorous and indicative of the broad range of definitions/interpretations that can exist with regard to a "single" concept (in his case, derivative; in yours, continuous; in others, fraction, etc). – Benjamin Dickman Feb 19 '16 at 23:04
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    Additional motivation of the question: The paper What really is a continuous function? by J.F. Harper demonstrates that in the beginning 20th century it wasn't easy to find the right formal definition for continuity. Quote from the paper: "Eleven experts on analysis defined continuity in ways that give five or six different results, two of the authors committed an error, and one famous book does not resolve an ambiguity." – Stephan Kulla Feb 20 '16 at 0:13
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    For teaching, you might try to put yourself in the position of your students: Which of these do you think your students would feel is most worthy of study (or better, which do you think YOU are best able to argue is a property worthy of study)? A couple of hundred years ago numerically solving equations by making use of the intermediate value property was very much front and center of student's mathematical training before calculus (which was most of their undergraduate career), but I suspect this is not the case today, so #1 might not be all that relevant in view of their past experience. – Dave L Renfro Feb 22 '16 at 20:45
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    I beieve @tampis' comment is right on the money. The concepts of "limit" and "continuity" are fundamental in calculus/analysis, but are also quite subtle (it took more than 2000 years from Archimedes' use of limits to get a rigorous definition of the term, it took hundreds of years from suspecting continuity was important to getting a satisfactory definition some 60 years back). To believe they will be easy to grasp in their multiple ramifications by complete newbies is just ludicrous. – vonbrand Feb 24 '16 at 13:11
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    Relevant: What does continuity in general mean?. – dtldarek Mar 4 '16 at 23:25
up vote 9 down vote accepted

Have a look at the paper written by Nunez et all:

EMBODIED COGNITION AS GROUNDING FOR SITUATEDNESS AND CONTEXT IN MATHEMATICS EDUCATION.

In essence, they argue that it is better to be causious if you want to "motivate the formal definition of continuity starting from the intuition" you have suggested in your question. In the following passage, "natural continuity" refers to drawing without lifting the pen.

For the purposes of this article, the pedagogical problem can be summarized as follows: students are introduced to natural continuity using concepts, ideas, and examples which draw on inferential patterns sustained by the natural human conceptual system. Then, they are introduced to another concept – Cauchy-Weierstrass continuity – that rests upon radically different cognitive contents (although not necessarily more complex). These contents draw on different inferential structures and different entailments that conflict with those from the previous idea. The problem is that students are never told that the new definition is actually a completely different human-embodied idea. Worse, they are told that the new definition captures the essence of the old idea, which, by virtue of being ‘intuitive’ and vague, is to be avoided.

If I had more time, I would summarize their argument. For the time being, I make this a community wiki to invite other people to complete this answer.

The prototypical way for a function to not be continuous is that of a jump discontinuity. Imagine a jump discontinuity on the order of a few micrometers, like the width of a hair. If you are tracing the graph of the function with an everyday pencil, you would slide right across the discontinuity without even noticing its presence. However, if you shrunk yourself and your pencil to the micrometer scale, you would suddenly notice the discontinuity. So the width of the pencil corresponds to $\epsilon$. The parameter $\delta$ is a localizing parameter that allows one to focus on the region near the discontinuity. There might be another location in the domain where there is a jump discontinuity on the order of a few angstroms, so the localizing parameter $\delta$ might need to be smaller. In order to rule out a jump discontinuity at $x_0$ you have to look at all small scales to make sure you are not overlooking anything, hence for all $\epsilon>0\ldots$.

So I think a natural way to develop the informal definition of continuity to the formal definition is to focus on the intuitive negation of continuity in the form of a jump discontinuity. But as you probably know, the formal definition of continuity permits some continuous functions that only baffle the mind when we try to visualize. For example continuous functions that are nowhere differentiable, or a function that is continuous only at one point.

There is an entirely different perspective on this entire problem which is revealed by asking: Is continuity what we should teach students? That is, before we think about motivating a formal definition of continuity, we might wish to question whether continuity is the concept that we really need students to know.

An alternate perspective is that the concept which we want to motivate is actually uniform continuity, and that we can approach this concept using much more concrete methods than with the current conception of continuity at a point.

The motivation for uniform continuity comes from calculating values of functions (like the square root function $\sqrt{x}$) to a certain number of decimal places and finding that if we have an approximation to say 2 decimal places then this approximation is constant over an interval of arguments. Another way of saying this is that a function is uniformly continuous if we expect that its decimal representation "settles down" over a certain interval of arguments. Said yet another way: we do not give a function arguments which are not rational i.e. we do not give endless decimals to a function when we're calculating, rather we suppose that a function giving an endless decimal is constant up to a certain number of decimal places over some finite interval. There can be some complexity in the fact that an infinite decimal number can have more than one representation e.g. 1.999... is a synonym for 2.0000... ) but these can be addressed just as concretely as the motivation for uniform continuity.

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    The problem is then that functions such as exponentials are not uniformly continuous, and one is lead to use phrases such as "uniformly continuous on bounded sets" which may not be the clearer way to state hypotheses. Think about defining the natural logarithm as the reciprocal of the exponential: this leads to subtleties that may contain germs of confusion for students, compared to the notion of continuity. – Benoît Kloeckner Mar 7 '16 at 15:42
  • What I have described is in fact uniform continuity on a bounded interval: it does not introduce any more "germs of confusion" than might be introduced by an attempt to introduce continuity at a point. I mean something specific by this claim: students are likely to think of continuity AS uniform continuity over a bounded interval, and everything relevant to a student's intuition starts from that belief. – John Mar 7 '16 at 21:40
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    I take your point, and I am in fact somewhat sympathetic to your approach (the physical meaning of continuity is prime for me, and physically if you want to say anything relevant you need an explicit modulus of continuity, hence uniform continuity). However I feel your claim that "students are likely to think of continuity AS uniform continuity over a bounded interval" would need to be substantiated before being stated so strongly. – Benoît Kloeckner Mar 8 '16 at 8:22

I believe it can be easier, at first, to look at the formal definition of continuity in terms of function application commuting with limits, which says that $\lim_{x \to A} f(x)$ is the same as $f(A)$. The explanation is more clear if also you write $f(A)$ as $f(\lim_{x \to A} x)$.

Thus:

  • $f(A) = f(\lim_{x \to A})$ is "where you are" at at time $A$
  • $\lim_{x \to A} f(x)$ is "where you should be" at time $A$, based on where you were at nearby moments of time.

If someone else naturally followed the function, then at time $A$ they ought to arrive at $\lim_{x \to A} f(A)$. But this means that if $f(A)$ is not the same as $\lim_{x \to A} f(A)$, then the actual function would require "lifting a pen", in some informal sense.

Of course, as with any explanation, this requires the students to apply some amount of intuition. And it makes more sense if we already know that the function is continuous everywhere else, so that it is possible to trace "the rest" of the function. But, if students are already able to visualize limits, this helps them visualize continuity.

Also, because the limit form of the definition of continuity is vital for applying continuity to compute limits in the context of calculus, emphasizing the link between continuity and limits can have other benefits, compared to the $\epsilon-\delta$ definition.

There is something to be learned from the history of the concept. The modern concept of continuity was introduced by Cauchy in 1821. This concept is probably best explained in two stages:

(1) Cauchy's definition of continuity as infinitesimal $x$-increment always producing an infinitesimal change in $y$;

(2) once students understand the basic intuition, the epsilon-delta paraphrase can be explained in terms of trough and trap.

In more detail, in order for the value of the function to end up in a trap of a certain size, we must "feed" it values from a trough that's sufficiently small.

For example, the absolute value function $|\;|$ is continuous at the origin because for every infinitesimal $\Delta x$, the corresponding $\Delta y=|\Delta x|$ is similarly infinitesimal.

  • How would you motivate Cauchy's definition from the intuition that the graph can be drawn without lifting? Would you directly start with Cauchy's definition or would you motivate it? – Stephan Kulla Feb 21 '16 at 16:08
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    We taught infinitesimal calculus to 130 freshmen this year and this is exactly what we did. I just finished grading the final and the mean is about 75. They did fine! – Mikhail Katz Feb 21 '16 at 16:12
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    If you use infinitesimals dx and dy to frame continuity, then you end up with a construct where continuous implies differentiable. After all, with dx and dy you can get dy/dx. So this is not a good approach to get to the formal definition of continuity. – user52817 Feb 21 '16 at 19:24
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    @user52817, the absolute value function is continuous in Cauchy's sense, but not differentiable. Why would continuity imply differentiability? Do you really believe Cauchy was an imbecile? – Mikhail Katz Mar 2 '16 at 9:38
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    @user52817 This is an error in Kline which he copied from Boyer. I suggest you look up Cauchy in the original instead of relying on compromised sources. – Mikhail Katz Mar 2 '16 at 16:06

To give a partial answer to my question: In "A radical approach to real analysis" David Bressoud gives a good explanation why the intermediate value property (IVP) is no good choice for continuity (pp. 91 ff):

  1. The image of a closed interval might not be bounded.
  2. The sum of two functions with the IVP might not be a function with the IVP.

For (1) take $f:[0,1]\to\mathbb R: x\mapsto \begin{cases} \tfrac 1x \sin\left(\tfrac 1x\right) & ;x\neq 0 \\ 0 & ;x=0 \end{cases}$

For (2) take $f:\mathbb R\to\mathbb R: x\mapsto \begin{cases} \sin^2\left(\tfrac 1x\right) & ;x\neq 0 \\ 0 & ;x=0 \end{cases}$ and $g:\mathbb R\to\mathbb R: x\mapsto \begin{cases} \cos^2\left(\tfrac 1x\right) & ;x\neq 0 \\ 0 & ;x=0 \end{cases}$ so that $f(x)+g(x)=\begin{cases} 1 & ;x\neq 0 \\ 0 & ;x=0 \end{cases}$

  • Unless I am misunderstanding your intention, I think "must" should be "might". It is certainly NOT true that (1) and (2) must not be true for IVP functions. For example, both (1) and (2) are true for the functions $f(x) = g(x) = x$ and these functions have the IVP. – Dave L Renfro Feb 22 '16 at 20:40
  • @DaveLRenfro: Thanks a lot! I corrected my typo... – Stephan Kulla Feb 23 '16 at 19:31
  • Incidentally, regarding your (1) and (2): There exist Darboux functions such that the image of EVERY closed interval (of positive length) is ${\mathbb R}$ (see here). In 1929 Adolf Lindenbaum stated (without proof in a conference abstract) that ANY function can be written as the sum of two Darboux functions. I don't know off hand who first published a proof, but I know that proofs were published Solomon Marcus (1960), Wacław Sierpiński (1963), and Paul Erdos (1964). – Dave L Renfro Feb 23 '16 at 20:03
  • Do you mean sin(1/x)^2 + cos(1/x)^2 = 1 ? – j4n bur53 Aug 26 '17 at 12:44
  • @j4nbur53 Yes, of course. :-) Thanks, I've corrected the answer – Stephan Kulla Aug 30 '17 at 20:25

Property (3), i.e., the $\varepsilon-\delta$ definition of continuity, has numerous motivations/interpretations. For example, continuity can be interpreted as accuracy. Suppose we are shooting a cannon located at the top of a hill. Even when the mathematics is perfect, our limited knowledge of the many parameters that affect the accuracy of our shot will make us under- or overestimate where the cannonball hits the ground. Let's take just one: the accuracy of our measurement of the height of the hill. If the height, $H$, varies, then so will the placement, $D$, of the shot. Of course, the smaller error in the former will lead to a smaller error in the latter and, furthermore, we can achieve any required degree of accuracy, $\varepsilon$, of our shot if we can ensure a sufficient accuracy, $\delta$, of the measurement of the height of the hill. In other words, the dependence of $D$ on $H$ is continuous. But if we replace this target shooting with a game of tennis, we will face discontinuity. enter image description here

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