I have just completed book AB1 from the final analysis block and I am 3/4 the way through TMA04. I feel that the clouds are beginning to lift and some way off, but now visible, is the finishing line! Perhaps this is an odd thing to say when the course is just 6 weeks old but after going through a difficult patch after working on M208 for nearly a year, it is a nice feeling when you begin to feel that things may be downhill from now on.
I didn't find the last part of GTB3 on the counting theorem too bad. The video that goes with the course was a big help and the examples that they discussed were interesting.
Well AB1 - yuk. The first couple of sections are ok on limits and asymptotic behaviour of functions, but then came the big elephant in the room, the epsilon-delta definition of continuity and the continuity of strange functions. I can't say that I really understand the epsilon-delta definition of continuity at all well yet. I managed the exercises and worked through the proofs, but I think a lot more practise is going to be required before I can master this stuff.
The strange functions are just a p in the a. I did wonder why it was necessary to prove that the Blancmange function B(x) was continuous when it is constructed from a sum of continuous functions. They showed that B(x) is convergent for each x in R by the Comparison Test, so why isn't this and the Sum Rule for continuous functions enough to say that B(x) is continuous? I don't know.
This may also sound odd but light relief has been to work through TMA04 on the first analysis block. Read a bit of AB1, feel totally baffled, and then for light entertainment find out if a sequence or series in TMA04 converges. Bliss in comparison.
It seems that I made the right decision when it comes to the TMAs. I decided not to agonise too much over my mathematical language and to trust my own instincts. This seems to have been justified as I have had the remainder of TM01 back from my tutor and he seems happy enough with what I have done. Although I argued successfully enough to score 100% on this TMA, there were one or two places where I didn't quite get to grips with the argument and overcomplicated things. I think this accurately reflects the topics that I will need to work on in revision. At least I didn't waste lots of time wondering whether I should be using the word 'hence' rather than 'it follows' or whether an implication should be an equivalence! Trust the gut!
Proving continuity or otherwise of the Blancmange function is the best bit of M208 :)
ReplyDeleteIt actually has relevance to the study of fractals as they are functions which is continuous but nowhere differentiable.
As for the epsilon delta definition of continuity keep practicing it forms the foundation of most analysis texts so if you want to continue your studies in that field you have no option.
In the topology course I'm doing it kicks off almost immediately with the epsilon delta defintion of continuity and I must have learnt something as I was able to answer the TMA question on it almost immediately.
So well done but keep sticking at the epsilon delta definition of continuity as it will set you up in good stead for further study in analysis.
I think I will have plenty of time for practise and perhaps some further investigation of the definitions. What did you think of my suggestion that the continuity of B(x) can be obtained from the fact that you are summing continuous functions and that the sum is convergent? Something like this must also apply to Fourier series.
ReplyDeleteLooking at the proof of the continuity of B(x) you can see how half of it is based on the continuity of the saw-tooth functions whilst the other half, the sum from N to infinity, is based on the upper bound of the sum. Sorry, this is a bit wishy-washy, but you will probably get my drift.
Another aspect of the discussion that seemed to be missing is what happens when a function is not continuous. How does the epsilon-delta definition break down? I can see how it happens myself, but there was no talk of it in the book.
To answer your question about why you can't just use the sum rule for continous functions the point is that that only applies at the same point. In the Blancmange function you are testing the continuity of each point at 2^n c which will be different for each value of n.
ReplyDeleteAs for discontinuous functions the definition breaks down at the discontinuity you will never find a delta such that the difference |f(x)-f(c)| < epsilon. So suppose you have a step function at x = 0 such that on the left x = 0 and x = 1 at the right then no matter how small you make delta episilon will always be a number between 0 and 1.
Hope this helps Chris
I think the Sum Rule does still apply here. Take, for example, the first two terms of B(x) which are s(x) and (1/2)s(2x). At the bottom of page 28, AB1, it says that the function s(2x) is a continuous function (on R, I think). Let g(x)=(1/2)s(2x), so g(x) is a continuous function by the Multiple Rule. Thus h(x)=s(x)+g(x) is continous at x=a, say, by the Sum Rule. You could then extend this argument by partial sums. I neach new partial sum you are adding another term of B(x).
DeleteTo see this geometrically you can imagine all the graphs of y=(1/2^n)s(2^nx) simultaneously. Choosing a point at x=a say, each graph is continuous at x=a, and so the partial sum is continuous. The issue arises as the partial sum is taken to its limit. Holes could appear in the graph of the infinite sum if the peaks and troughs of the teeth of the basic function "coincide" so that at a particular point B(x) tends to infinity or minus infinity.
However, this doesn't occur because B(x) is convergent for each x element of R by the Comparison Test (see page 27 of AB1). Hence this illustrates the point I am trying to make - why do we need the epsilon-delta definition of continuity to determine that the Blancmange function is continuous at x=a?
I agree with you about your second paragraph and is what I expected but it is a pity that they don't actually give this as an example in the book!!
This discussion is great by the way. I need to have these sort of discussions as they are far more interesting than just reading the text!
I have just found an interesting article in The Mathematical Gazette, Vol. 66, No. 435 (Mar., 1982), pp. 11-22. You can find the source in JSTOR via the OU library. It tries to give a pictorial view of the continuity of the Blancmange function.
DeleteHaving given it some more thought I think I can see that the potential problem is a bit like the problem of the continuity of sin(1/x) at x=0. With s(2^nx) the jagged teeth of the saw-tooth function become ever more frequent as n becomes larger and this could cause a discontinuity even though the function is bounded. Interesting!
DeleteOk let there be two functions f and g one continuous at x1 and the other continuous at x2 you cant use the sum rule of continuity to say that the sum of the functions is continuous at both x1 and x2. The sum rule as it is in Brannan states explicitly that if f and g are continuous at x1 then f and g are continuous at x1. It says nothing about whether f and g are continuous at x2.
ReplyDeleteIn the Blancmange function each partial sum is being tested for continuity at a different point so eg f2 is being tested for continuity at 2^2c and f3 is being tested for continuity at 2^3c and so forth. Sorry if that just repeats what I said in the first point. But I think you really have missed what I was trying to say.
We should discuss this over a pint or two. Maybe after the 23rd I really think it's worth understanding this proof and the consequent proof of the non differentiability of the function properly as it is only then that the power of the epsilon definition of continuity comes into it's own.
Best wishes Chris
Sorry I think I missed your point however I'm not sure you can just say that because the functions are continuous for all points at R therefore the sum of two different functions are continuous at different points of R. It seems to go against the spirit of the epsilon delta definition of continuity which this example is trying to illustrate. Your point seems to assume continuity which is what the argument is trying to prove.
ReplyDeleteHi Chris, I think you are misunderstanding me still. I am not picking different points in R, I am picking the same point in R, to determine the continuity of the partial sum. Each term of B(x), namely (1/2^n)s(2^nx), is a different function, call them g1(x), g2(x),... Each function gn(x) is continuous and by the Sum Rule the partial sum is continuous at x=a, say.
DeleteI am not assuming continuity, I am using the rules for continuous functions given in the text. I have absolute confidence that the partial sum is continuous as I have described. What I am not sure is whether it is in the limit as the partial sum tends to an infinite sum. I willing to bet, though, with certain conditions, you could prove that this was so!
Hi having looked again at the proof again in AB1 it's not a question of 'why do need the epsilon delta definition of continuity' It's more a question of using it to guarantee that each partial sum will be less than 1/4 epsilon as the top of page 28 states.
ReplyDeleteHave to admit the proof in Brannan is a bit more clearer on the details
As you say it's good to debate this stuff.
If you do Topology you will be relieved to learn that epsilon and delta disappear to be replaced by mappings between inverses of open sets. However as many of my books on Applied mathematics and Differential equations use epsilon delta type proofs all the time, it definitely is worth getting to know and understand these tpyes of proofs, even though the TMA and exam questions on this are fairly routine and probably scratch the surface.
And just so you are reassured I'm still not a robot :)
I am not trying to get out of using the epsilon-delta definition. I think it is very interesting. It is a bit like trying to draw a box round the graph of the function and making sure that all of its wiggles fit into the box.
DeleteThe proof in AB1 does use the continuity of the partial sum, as you say, at the top of page 29. The other bit of the proof, summing from N to infinity is the interesting bit and this approach is probably needed to show that the infinite sum behaves properly! Heck, I am sounding like a physicist again!
yes I think part of the confusion is the different treatment of each bit which is why discussing these things face to face is such a good idea. I'm still not sure why you think
ReplyDelete2^nc is the same point as 2^(n+1)c which are different points and where each term of the partial sum is being evaluated.
Seems a pint is in order then :-)
Delete