|Credits: 10||Convenor: Will Light||Semester: 2|
|Assessment:||Continual assessment: 20%||1.5 hour exam in January: 80%|
Let's start with continuity. Suppose we have a function f mapping some sort
objects (which we will call `points') to numbers. What does it mean for the
function to be continuous at the point p? Well, roughly speaking, if q is
close to p, then f(q) must be close to f(p). OK so far? Well, not really.
We've already been much too loose in the way we have formulated the problem.
First off, we described the things on which f acts as `objects', and then
the more emotive word `points'. Then we went on to talk about these `objects'
`points' being close together. This places some restriction on the sort of
objects allowed. For example, our set of objects could consist of all cheeses
currently lying on the shelves of Sainsbury's supermarket. For a particular
cheese p, f(p) (which remember has to be a real number) could be the weight
of the cheese. Now I think you're going to have a hard job coming up with an
appropriate definition of two cheeses being close together! Note here that the
problem in determining a sensible meaning for `closeness' only occurs in the
input space, or the domain of the function - the set of all cheeses in
Sainsbury's. Closeness in the output space is easy to define isn't it? We would
say that f(p) is close to f(q) if |f(p)-f(q)| is small. But hang on a
minute. Even this idea is not precise enough for a mathematician (or even for
an applied scientist, or person working is some sort of technological
If I was to discover that the north wall of my house was 2mm higher than the
south wall, I wouldn't exactly be rushing to sue the builder for negligence. On
the other hand, if the garage sets the gap between my spark plug electrodes
an error of 2mm, then I would be bleating in their ears that something is wrong
with my car. So the notion of closeness needs to be expressed a lot more
carefully. (The formalisation of the concept of closeness in leads
inexorably via MC146 to the formal definition of continuity using
and technology.) But around the turn of the century,
mathematicians wanted to get away from the idea of `closeness'. This desire was
bound up with another very sophisticated development in thinking, which has
today become second nature to professional mathematicians. Traditionally,
functions used to be things which mapped numbers to numbers, or more generally,
`points' in to `points' in . But it became increasingly common
think of functions in a much more abstract way. In the vanguard of this
development was the idea that the so-called `points' could themselves be
functions. For example, the `points' could consist of all real-valued functions
differentiable on the whole real line. Each `point' is now a function from
to . Thus the function defined by the rule
f(x)=x2 would be a `point'. Now consider the mapping called
`differentiation'. Applying this mapping to the `point' f gives the new
function defined by the rule g(x)=2x. Mathematicians
wanted to study such types of mappings or functions acting on
`points' which are themselves very sophisticated objects. A major force in this
movement was the French mathematician Maurice Fréchet (1878-1973). The work
Fréchet and his contemporaries has had a profound effect on the way we teach
you! In MC144, you first met the idea of a function. It probably seemed then
pretty weird, because all you were given were two abstract sets A and B and
the rule for getting from A to B, often called f. Nobody encouraged you
believe that either A or B were made up of numbers. Instead, we tried to
convince you that A and B could be pretty much anything you liked: cheeses
from Sainsbury's, functions, points in . However, because we are
sensitive to the difficulty of this abstraction, we often used examples in
A and B were both (subsets of) . OK, let's hope you see that working
with pretty abstract objects is part of modern mathematics, and let's return to
our idea of continuity. My previous arguments have been designed to show that
need some idea of `closeness' before we can begin to make sense of continuity.
In the example I have already introduced, where the `points' were themselves
functions from to , it is not too difficult to come up with a
sensible notion of when two `points' (functions) are close. MC248 takes this
route, but we are bound for higher things! The fundamental question we shall
answer in the first lecture of the course is
Of course, we cannot in this course be anarchists, and develop our own notion which fails to coincide with the MC146 notion of continuity. So there are constraints.
Let's go on now to talk about the other great concept of integration. At the outset, the `A'-level student knows that integration has two meanings - definite integration (which is the area under the curve), and indefinite integration (which is the reverse process to differentiation). Of course, the two are linked through the fundamental theorem of calculus. On a personal note, it's one of the great surprises to me, and something that I find absolutely fascinating, that the area under a curve f between a and b can be evaluated if only you know a function g such that g'=f. One version of the fundamental theorem is then that
Magic! But in this course, integration means definite integration, that is something to do with area under curves. Have you noticed we have switched back to functions ? Without this switch the above discussion would not make sense. Also at this stage in your undergraduate career, all you have experienced is the Riemann integral. The upper and lower sums idea was introduced in a paper by Riemann in 1854. However, it soon became plain that Riemann's definition had some flaws. The work of Cantor (1874) had lead to the concept of the size of a set. Two sets have the same size if a 1-1 correspondence exists between them. Particular importance is attached to sets with the same size as the integers. Such sets are called countable. Cantor showed that the rational numbers are countable but the real numbers are not. Thus there are `as many' rationals as integers, but there are `far more' reals than integers.) What frustrated practitioners of the art of integration was that the function f which is zero at each rational value and 1 at all other values is, according to Cantor, 1 `most of the time'. It therefore ought to have for all values . (By the way, in olden days the idea of integrating a function which had an absolutely huge (i.e., infinite) set of discontinuities was a major pastime.) What frustrated the masters of integration was that according to Riemann's definition, this function was not integrable. (It's not too hard to show this - look up the definition from MC248 and have a go!) So the Riemann integral had flaws, which were not really resolved until the thesis of Henri Lebesgue (1875-1941). Following Lebesgue, much the same programme was put in place for integration as for continuity:
The ability to apply taught concepts to new situations.
The ability to write mathematics concisely.
The ability to construct logical arguments.
The ability to carry out elementary manipulations with unions and intersections of sets.
W. Rudin, Real and Complex Analysis, McGraw Hill, 1970.
W. A. Sutherland, Introduction to metric and topological spaces, Oxford : Clarendon Press, 1975.
W. Light, Introduction to Abstract Analysis, Chapman and Hall, 1990.