I have now completed the last of the books in the first analysis block of M208. This book was all about continuity in real functions, and as for the other analysis books, I really enjoyed working on it. In a nutshell, to determine whether a function f(x) is continuous at x=a, you have to ensure that, as x tends to a, f(x) tends to f(a). Of course, you need to consider the approach to x=a from both smaller and larger values of x.
I am already familiar with the idea of continuity having come across it in previous maths studies but this course deals with the subject in a much more rigorous way, using sequences in x to determine how f(x) behaves as x tends to a. One gripe that I have is that a major plank of the rest of the book, the Intermediate Value Theorem, is only proved for a special case and not proved in general. When a lot of M208 is devoted to carefully proving theorems it is sometimes surprising to find omissions, but I suppose that in some cases the proofs are too involved to reproduce for the course. The Intermediate Value Theorem is the basis of how we can define inverse functions for increasing or decreasing continuous functions.
Another gripe that I have is that it is sometimes difficult to judge the level of detail that is required in some of the answers to the exercises. Sometimes I find myself putting in too much detail and at others too little. Take Ex. 4.1 on p38 of AA4, for example. The solution says that f(n)=n2-1/n tends to infinity as n tends to infinity by the Reciprocal Rule. Is it really necessary to quote the Reciprocal Rule at this point? Isn't this obvious enough? In the examples in the text they manage fine without quoting this rule, so why suddenly do so here? There has to be some point at which you don't have to quote every theorem and rule that you have learned in M208 in order to prove something otherwise it is going to be extremely tedious/arduous. In fact, the solutions to the exercises are meant to be a guide as to how questions should be answered, but some rules, strategies and theorems are omitted in these solutions when convenient. So how are we to judge this?
Hi Duncan.
ReplyDeleteHaving taken M208 in 2010 (and I'm sure Chris will confirm this), you need to go rigour-mad on the TMAs to get all the marks.
If there's a result by some rule, you'll need to state this.
While it's a bit of a pain at the time, it does stand you in good stead for further study as this method of justification becomes second nature.
I remember receiving red pen back for using => rather than <=> on one TMA, and getting comments about the use of "thus" against "it follows" (or was it the other way round?)
Oh no, you're joking (but probably not). I particularly hate having to choose between => and <=>. I was hoping to use => a lot, because using <=> means having to ensure that both directions are true. If it goes to the level of "thus" versus "it follows" then I will slit my wrists. The problem is that if I thought that the course material was as exact as all that I would follow their lead, but they are not, and that is my point. Maybe I will just bore them silly with such an unbearable level of justification that they will get bored of correcting it. The other thing is that I have found quite a few typos and a few errors in the text and so if they bore me with thus vs hence I will ensure that I go through every mistake in the text with the tutor and ensure they have to issue a whole raft of errata.
ReplyDeleteI'm afraid dear Alan insists on <=> especially in inequalities so best get used to it :)
ReplyDeleteDuncan, go for it. It's a good laugh when you get well into it.
ReplyDeleteI used this level of rigour in MST209 this year and it stood me in good stead.
All the best.