The history of pointwise/uniform convergence has an interesting and pertinent history. There is a good account of the history in Appendix I of Lakatos, I. "Proofs and Refutations". Broadly speaking, Cauchy published a proof that the limit of a sequence of continuous functions must be continuous. Abel pointed out that this proof "admits exceptions" (Fourier series were known) and the distinction between pointwise and uniform was teased out by Seidel and clinched by Weierstrass. Bob Burn University Fellow, Exeter University Sunnyside Barrack Road Exeter EX2 6AB 01392-430028 ________________________________________ From: Post-calculus mathematics education [MATHEDU@JISCMAIL.AC.UK] On Behalf Of Jonathan Groves [JGroves@KAPLAN.EDU] Sent: 30 January 2010 20:22 To: MATHEDU@JISCMAIL.AC.UK Subject: Re: Uniform convergence and pointwise convergence
Joel Feinstein wrote:
> I have now combined the three screencasts on > sequences of functions into one 80-minute movie, > available from > http://wirksworthii.nottingham.ac.uk/webcast/maths/G12 > MAN-09-10/Chapter9/ > I am experimenting a little with my webcam: this > combined movie includes three short sections where I > am talking to the webcam, and warning you about the > various incidents that affected the recordings! > I am interested in whether it is important to include > footage of the teacher to make screencasts more > lively. See my blog entry on this at > http://wp.me/posHB-7l > I would welcome comments on this: I have the facility > to record two streams simultaneously. and combine > them manually (probably impractical) or in a variety > of automatic ways. > Joel Feinstein > http://explainingmaths.wordpress.com/
Joel, I will make more comments later about the footage of the teacher being included in the videos when I watch your latest video lecture with you included in the video. I would think that something like that can help make videos more lively since the students are probably used to seeing the lecturer as opposed to listening to lectures with the lecturer being invisible. I myself have attended more math talks and lectures with the lecturer seen than math talks with the lecturer unseen.
I did see your video on pointwise and uniform convergence of sequences of functions with all sessions combined. I do think the examples and pictures do effectively illustrate the difference between pointwise and uniform convergence of sequences of functions. I myself would probably try to include an additional example of uniform convergence, but maybe you were short on time for that. There is often not enough time to do all the examples we would like to do. I face that problem a lot myself and often don't have enough time to explain everything I want to explain.
You did explain in your video one reason we do discuss pointwise convergence of a sequence of functions: It's easier to understand than uniform convergence, so it can be a good way to introduce the notion of a sequence of functions and the easiest way to try to define what convergence means for such a sequence.
I think we mention pointwise convergence for another reason: It seems to be a natural way to define convergence of a sequence of functions. At first glance, nothing seems wrong with it. But then we discuss examples of sequences of continuous functions whose pointwise limit is discontinuous or sequences of unbounded functions whose pointwise limit is bounded to show that this innocent-looking definition leads to all kinds of problems. I will have to check into the history of analysis for more information, but I would guess that pointwise convergence was the first attempt by mathematicians to define what it means for a sequence of functions to converge. Even if I am wrong about the history, this is still a good example that shows that we must choose our definitions carefully; otherwise, they might not behave like we want them to.
Showing that uniform convergence is superior to pointwise convergence by establishing all these properties does take a lot of work, especially for the students who haven't gotten used to these concepts yet, but it is important to do so. However, I can see a way at the beginning to explain why, and I think this explanation is easy to understand. Establishing these properties later then gives further confirmation that this explanation does hold: Uniform convergence guarantees that as n gets very large, each individual function f_n is a good approximation of f for all x. If convergence is pointwise but not uniform, then f_n works well as an approximation of f for some x but not so well for other values of x. It is then clear that an approximation that works well for all x is much better than an approximation that works well for some x and not so well for other values of x. This is even more true if we have a sequence of approximations of f where the approximations eventually become excellent ones for all x as opposed to a sequence of approximations of f where none of them are excellent approximations for all x. The properties later established then give further confirmation that this explanation does indeed make sense.
Do you or any others think that this explanation at the beginning will help students see why we prefer uniform convergence over pointwise convergence though pointwise convergence is simpler to understand and appears to be a reasonable definition of convergence of a sequence of functions?
After seeing all these properties of uniform convergence that do not hold for pointwise convergence and after thinking about this explanation I just gave, I can see a lot of beauty in the definition of uniform convergence that is missing from the definition of pointwise convergence.
I had noticed that you had mentioned in your video that a sequence of bounded functions could converge pointwise to an unbounded function. I don't recall seeing any such example before, but I might have. I did try to construct such an example, and I believe I have found one:
Let D be the closed interval from 0 to infinity.
f_n(x) = x cos(x/n) for x less than or equal to n on D and f_n(x)=0 for x > n.
Then f_n(x) is bounded on D for all n. And f_n(x) converges pointwise on D to the function f(x) = x, which is unbounded.
I wonder if any of your students have found such an example.
You did mention that a sequence of unbounded functions could converge pointwise to a bounded function, but the first example f_n(x) = x/n works. I mention this only to let you know that I did notice you saying this as well and not just the other statement.