Re: Cantor Confusion

Actually it was David Marcus.

Sorry, sincere apologies, too many Davids.


That function cannot be made continuous at 0 at all. It is an essential
discontinuity.

Doubtless you are right, it is very badly behaved, but I would like to know the basis for the assertion.

We can't define "the function and all its derivatives" at x = 0. We can
define the function to be whatever is desired at x=0, but the derivatives
(if they exist at all) are completely determined by the definition of the
function. In particular, if the function is defined to be anything other
than 0 at x=0, then the derivatives do not exist. This is not a mistake;
this is mathematics.

Yes, I agree, you can't define the derivatives, only the function value at the point, my mistake. But I don't see the distinction between this situation and sin(pi/x) - in one case, the function can apparently be defined arbitrarily at x = 0, and in the other it has to be 0? If you define exp(-1/x^2) to be other than 0 at x = 0, then the derivatives will be discontinous (and infinite), and such a defined function will have no valid Taylor expansion, but so what? You are happy to introduce a discontinuity and an arbitrary value at x=0 on sin(pi/x) ?


I think that you lot discover meaningful mathematical
objects; I don't think you define them. If they are
universal truths, you could have an intelligent discussion
with the aliens when they land...

The underlying truths will be the same, but the definitions will almost
certainly differ.


Well that I totally agree with - it is the underlying truths that are important, not the language or formalism in which they are expressed. So, you agree that you discover (like a scientist), rather than define, mathematical objects of universal interest?

--
Andy Smith
.

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