Re: Cantor Confusion
- From: Dave Seaman <dseaman@xxxxxxxxxxxx>
- Date: Wed, 17 Jan 2007 00:08:47 +0000 (UTC)
On Tue, 16 Jan 2007 21:30:11 GMT, Andy Smith wrote:
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.
In order for a function to be continuous at a point x=a, three things
have to happen:
(1) f(a) must exist.
(2) lim_{x->a} f(x) must exist.
(3) The values in (1) and (2) must agree.
When (2) fails, as is the case for sin(pi/x) at x=0, it means that no
assignment of f(a) can possibly make the function continuous there.
That's called an essential discontinuity.
By contrast, if (2) holds but (1) or (3) fails, then the function can be
made continuous by a suitable assignment of f(a). That's called a
removable discontinuity.
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?
There is no distinction between the two cases. Each of those functions
can be defined arbitrarily at 0, or left undefined there.
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) ?
If a function is not defined and continuous at x=0, then the derivatives
do not exist at all at that point. There is certainly nothing wrong with
that. It makes no sense to talk about derivatives being discontinuous or
infinite if they are simply undefined.
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?
Discovery and definition are not mutually exclusive. Definitions provide
us with a convenient way to state our discoveries. With a different set
of definitions, the same discovery might be worded quite differently. I
don't mean in the sense that English and French are different;
English-speaking and French-speaking mathematicians tend to use mostly
the same definitions in their respective languages. A mathematician from
Tau Ceti might very well choose different fundamental concepts to assign
names to, and therefore the statement of our familiar theorems might come
out seeming quite different.
--
Dave Seaman
U.S. Court of Appeals to review three issues
concerning case of Mumia Abu-Jamal.
<http://www.mumia2000.org/>
.
- References:
- Re: Cantor Confusion
- From: David Marcus
- Re: Cantor Confusion
- From: Andy Smith
- Re: Cantor Confusion
- From: Dave Seaman
- Re: Cantor Confusion
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