Date published: 2005/02/14

Tom Körner gave a CMS (Centre for Mathematical Sciences) Colloquium today on why analysis is so hard for students (even Analysis 1, the first year course). The problem is that there are lots of epsilons and deltas thrown around and it is easy to get lost. But are the difficulties artificial in that if the functions are "well-behaved" everything gets easier? Are analysis courses hard because they are taught by analysists?

For his main example, he looked at the second mixed partial derivative of a function, f, and asked why does d2f/dxdy = d2f/dydx (all the derivatives partial) for "well-behaved" f. (Apparently Euler was the first to notice it and Weierstrass the first to prove it.). He gave the following possible (facetious) answers:

- It is not true because there is an example which is regularly set in the (Cambridge Maths) Tripos to show it is not true. (Of course this is for functions which are not "well-behaved".)
- The "well-behaved" condition includes the statement that we can interchange the limits in the definition of the second derivatives, and from that the proof is straightforward. (But the problem here is why is it obvious you can interchange the limits.)
- We once saw a proof with lots of epsilons and deltas in it.
- "Well-behaved" means that f is like a polynomial and the result is true for polynomials.
- Repeated experiment (for various f) shows the result is true.

The fifth point above means that perhaps we find it hard to think of unusual functions so all examples will almost by default satisfy the result. He mentioned a theorem by Kolmogorov and Arnol'd which shows that any continuous function of two variables can be written in terms of functions of one variable and addition. So in some sense there are no unusual continuous functions of two variables. However Vitushkin apparently proved that the Kolmogorov and Arnol'd theorem does not hold for continuously differentiable functions.

Most of the rest of the lecture was used to show how our intuition about theorems on the real line often does not work when we restrict to the rationals. For example, the Intermediate Value Theorem and Mean Value Theorem do not work. For the Intermediate Value Theorem for a counterexample you can use the function f(x) = 1 for x2 < 2 and f(x) = -1 for x2 > 2. For the Mean Value Theorem for a counterexample you can use the function f(x) = x for x2 < 2 and f(x) = x-4 for x2 > 2. The basic point is that the "discontinuity" does not occur because it would have to be at x = sqrt(2), which is not rational. He also showed that polynomial approximation does not work with rationals in the same way as it does with reals for continuously differentiable functions.

So part of the explanation of why things were difficult to prove was that in the real context they could be true and in the rational context they could be false, and "the rationals are indistinguishable from the reals" so people do not have great intuition about these things ("people say the rationals have holes but have you ever seen one?"). And "nobody has ever seen an irrational number" (well, in some sense that is not true since, for example, as is well known, an isoceles right triangle with sides 1 has hypotenuse sqrt(2)).

The bottom line seemed to be that analysis is hard because analysis is hard.

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