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What could be a good "layman" metaphor for illustrating the difference between uniform and pointwise convergence of function series? I am teaching calculus to engineering undergrads; for many of them, the definitions are way too abstract, and mathematical examples are not that illuminating.

I was thinking about people "converging uniformly" towards the end of their kindergarten time (in at the age of 3, all out at the age of 7, say) and "converging pointwise" towards the graduation from university (some need 5 years, some need 10...). However, it turns out that life is not that simple even in the kindergarten, at least in the country where my university is, and I am looking for a more precise metaphor.

EDIT

Most of the answers show that my question is a bit too vague. It is definitely not my intention to substitute a mathematical definition by any kind of metaphor, or to start the discussion in the class with such a metaphor, or to use it repeatedly or systematically. I was wondering what can I say en passant when the definitions are already given, the examples are discussed, and this is the third class about uniform convergence, but some people still stare at me with empty faces. For those people, the mathematically rigorous presentation obviously did not work, and they need something that might motivate them to think over once again.

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My advice is generally to stick to the rigorous definition with plentiful examples of proofs for sequences of functions that converge one way or the other, and skip metaphors all together.

If you have to have a metaphor, you might consider runners in a race, each confined to a lane. The runners all move at different speeds, some perhaps even slowing as they approach the finish line, some very far away or very close at any given time. The runners in each lane need have nothing in common- some runners might even be approaching the finish line from the other direction. Some finish at different times than others, but they all finish at the finish line.

This race is like point-wise convergence, the runner's lane position is like the value of the independent variable, and the position of each runner is like the values of the intermediate functions at each value of the independent variable as they approach the respective value of the limiting function- the finish line.

Uniform convergence, on the other hand, is like all of the runners moving from their starting positions to the the finish line in roughly the same amount of time. Since they are moving in a coordinated fashion, the amount of the race that they have finished is not different from lane to lane, it only depends on how long the race has gone on.

But this metaphor still has some problems, because it's a metaphor. Instead of using a metaphor, though, I recommend that you simply provide ample examples and explain that showing point-wise convergence is really demonstrating that for every number $x_0$ in some interval, the sequence of numbers $(f_n(x_0))$ converges to the number $f(x_0)$, while uniform convergence is really a statement that the series of functions $(f_n)$ converges to a function $f$ on some whole interval.

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    $\begingroup$ Agreed. Especially if you have plenty of pictures. $\endgroup$ – Pat Devlin Dec 15 '16 at 12:54
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    $\begingroup$ Last paragraph nailed it, for me. Thanks. $\endgroup$ – Alecos Papadopoulos Dec 18 '16 at 21:00
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I don't think you need a metaphor at all — their intuition of convergence is probably already about uniform convergence, you just need:

  • To give surprising examples of pointwise convergence, to motivate the need for the more refined notion
  • To give them the algebra they need to explain how uniform convergence is different.

One example of a surprising sequence is that

$$ f_n(x) = \begin{cases} 0 & x < n \\ 1 & n \leq x \leq n+1 \\ 0 & n+1 < x \end{cases} $$

converges pointwise to the zero function, despite the fact the function doesn't "visually" get smaller at all. We can do this with continuous functions and without going off to infinity as well:

$$ g_n(x) = \begin{cases} 0 & x \leq 0 \\ n x & 0 \leq x \leq \frac{1}{n} \\ 2-nx & \frac{1}{n} \leq x \leq \frac{2}{n} \\ 0 & \frac{2}{n} \leq x \end{cases} $$

Multiply both examples by $n$ if you want something even more dramatic.

In both of these examples, you can put words to the idea that these 'ought' not converge by observing $\max_x |f_n(x) - 0| = 1$, which is clearly not decreasing to zero as $n \to \infty$. This translates to the characterization that $f_n \to f$ uniformly iff $$ \lim_{n \to \infty} \left( \sup_x | f_n(x) - f(x) | \right)$$ which makes it easy to see algebraically what's going on.

Then on the $\epsilon-\delta$ front, demonstrate how these bad examples of pointwise convergence all have $\delta$ depending on the choice of $x$


Depending on their particular engineering background, they may already know practical examples of this; e.g. if they've seen some sort of signal processing, they may know the Gibbs phenomenon.

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    $\begingroup$ The Gibbs phenomenon is exactly what I would have suggested. Forget metaphors and show actual examples of relevance to engineering settings. $\endgroup$ – KCd Dec 17 '16 at 10:24
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How about this: Think of the epsilon in the definition of either convergences as an allowable error. Now imagine you were given the task of filling 100 containers (of various sizes) with a certain prescribed amount (e.g., the $i$th container needs to contain $i^2$ Liters.)

Uniform Convergence: When the foreman comes around and checks your work, he will check all the containers at the same time and none of the containers can be off by more than 1 mL. It doesn't matter if that container has 1 L or 10000 L in it; none can be wrong by more than 1 mL.

Pointwise Convergence: When the foreman comes around and checks your work, none of the containers can be off by more than 1 mL, but if there are some that you didn't get close enough, he'll come back later and check again on those containers. As long as you eventually get them all within 1 mL, he's happy.

Analogy decoded: $f_t(i) \to f(i)$ where $f(x)$ is the desired amount of water in container $x$ and $f_t(x)$ is the actual amount of water in container $x$ at time $t$.

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I am not sure metaphors are as good a way to explain math as we tend to think, and I am quite convinced that metaphors that are assumed to be grounded in "real life" tend to have their silliness obscure their point. So let me try a go at a metaphor that is clearly not grounded in real life.

Sophia has an infinite collection of hourglasses, each of which has a duration of $n$ minutes (one hourglass for each positive integer $n$). She also has infinitely many arms, and use them to start all her hourglasses simultaneously. Then we know that each individual hourglass will be finished at some time, but at no time will all hourglasses be finished. In mathematical words, the hourglass all finish, but not in a uniform time.

Maybe the most interesting thing with this example is to look at the drawbacks it seems to exhibit.

First, the fact that the convergence is quite blunt, since each hourglass not only goes to the state "finished", but actually reaches it, is rather a simplification: it separates the concept of uniformity (which is pretty general) from the concept of convergence. You should also give other, mathematical examples of uniformity and non-uniformity (it all boils down to inversing quantifiers, and their I like the metaphor: "each girl is liked by some boy, but no boy likes all girls").

Second, having an infinite number of hourglasses may seem a bit too much. But it is necessary, since finiteness would make uniformity automatic! This is obvious by taking the maximum duration of an hourglasses, and can be used to prove that the convergence of a vector-valued sequence reduces to convergence of each coordinate individually. This is interesting, because the context in which non-uniform convergence occurs, function spaces, is precisely the infinite-dimensional case, where the max argument fails.

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Other answers suggest sticking to the definitions and just teasing out their logic carefully. I agree that this is a good method and you should mostly stick to this. @Behzad's answer with the picture and the notion of a "tube" around the limiting function's graph is also a good one.

I'm going to suggest an analogy that you might consider using, in addition to the other suggestions. I don't think it's perfect, and I don't believe anything could be better than simply working through all the details of the formal definitions. However, when you introduce the concept, you may want briefly mention this analogy, and also draw some "tubes" on the board. This way, students at least have some visual intuitions to latch onto when you start diving into the formal epsilontics of the definitions.

Think about snow falling onto flat ground. (That is, the limiting function is the constant 0 function.) Imagine a sheet of snow sliding off the roof and falling into your yard. This is sorta like uniform convergence: all of the snow is approaching the ground at the same pace, even it's not at the same height all the time. If you put your eye level at a certain height above the ground, there must be some point in time after which all of the snow is below your eye level. Some of the snow might be lower than other parts, but surely there won't be some snow that is lagging so far behind the other parts that it's not close to the ground yet.

Same situation, but instead think about individual snowflakes falling from the sky to cover the ground. This is sorta like pointwise convergence. You can be assured that every single individual flake will eventually reach ground level. However, each one is falling at its own pace. While some have already reached the ground, some may have just left their clouds and are well above the ground. There cannot be a single point in time after which all of the flakes are below your eye level. You have to track each individual flake separately.

Again, I'm aware this is not perfect and I'm sure you could pick apart the analogy. However, I think it may be helpful to start with a brief mention of this idea, then spend time working with the formal definitions and some examples, and even then returning to this analogy and asking the students, "In what ways was the analogy reasonably correct, and in what ways was it flawed?" If they are able to have a meaningful discussion about not only the original concept but also an analogy meant to illustrate that concept ... well, I think that means they learned something!

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    $\begingroup$ I quite like the idea (last paragraph) of circling back at the end to ask the students: In what ways was the analogy reasonably correct, and in what ways was it flawed? $\endgroup$ – Benjamin Dickman Dec 29 '16 at 3:52
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I dont think that we can explain any concept in mathematics dealing with infinitely many (like points, sequence) using real life examples, simply because in reality there is nothing infinite (an Avogadro number of particles, after all). The only way is to make assumptions which are unrealistic (as mentioned in a previous comment, like having infinitely many hands), and then achieve the desired goal.

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For uniform convergence: Consider a tube around the graph of $f$. No matter how thin the tube is, there exists an $N$ such that graphs of all functions (except possibly the first $N$ of them) completely fit in the tube.

In pointwise convergence there may be no such $N$ for some tube. enter image description here

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I actually am not sure that there is really any 'lay' way to think about this. But you could try to modify this to your preference:

I suppose you could compare a race of some kind with a security sweep. Even in a turtle race (more fun to watch than you think!) everyone might eventually get to the end, though at vastly different times. But if you are looking for a missing person everyone has to approach the end of the region being searched at roughly the same rate - and arrive together, or you might have missed something.

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From a non-mathematician (and let's hope I haven't made any elementary mathematical mistake):

They aspire to be engineers, so let's consider a new production process.

Being new, there exists a "learning curve", an initial period where we are not fully efficient in implementing it. Let $x \in (0,1]$ be the "intensity" with which we try to learn and implement the new process.We excluded $0$ from the domain to reflect the fact (or the belief) that we always pay even a little bit of attention. Let $A$ be full-efficiency output (for a given pre-specified level of inputs used). $n$ denotes the number of times the process runs a full circle. Our production function is

$$f_n(x) = \left (1-\frac{1}{2+nx}\right)\cdot A$$

The first term is the one representing the fact that we are gradually approaching full efficiency, based on the times we have run it ($n$), but also on the intensity with which we try to learn ($x$).

The production function converges pointwise to $f(x) = A$ for any given fixed $x \in (0,1]$. This is translated : with even a little but fixed level of learning intensity, we will eventually reach full efficiency output. That's comforting.

But the function does not converge uniformly because, say, if we set $x=1/n$ we get

$$\left |f_n(x) - f(x)\right | = \frac 13 A $$

that does not depend on the value of $n$.

DO WE CARE?
Does the difference between pointwise and uniform convergence reflects here something real-world important? It does: that we cannot rely on "automatic learning-by-doing" (the increase in $n$) to compensate for gradual loss ($x=1/n$) of conscious learning intensity, at least not for all rates of such gradual loss.

We examine the journey of a function looking at two possible influencing factors ($n$ and $x$). In pointwise convergence, we "commit" one of them to a specific value beforehand, so we examine separate scenarios of a function that is now influenced only by a single factor.

In uniform convergence, we allow both influencing factors to vary at the same time as we move along the sequence -and if uniform convergence obtains, we learn that even varying $x$ "in real time" does not eventually matter.

This, to me, is not a "subtle theoretical difference": it may be a very important difference, as I think my crude example above indicated. Introduce uncertainty about what the actual production function is, and the conventional wisdom that "sometimes it is better to commit rather than maintain flexibility" gets a good argument in favor (or equivalently, in an uncertain world, perhaps we may satisfy ourselves with pointwise convergence and accept its restrictive requirements as regards our behavior).

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