Re: Example for an assumption of the Implicit Function Theorem

From: Stephen A. Meigs (step314_at_aol.com)
Date: 07/13/04 

Date: Tue, 13 Jul 2004 01:44:53 -0400

"Paul Smith" <phhs80@hotpop.com> wrote in message
news:3e6c159c.0407121222.3b777f7b@posting.google.com...
> Dear All
>
> I am looking for an example to show that, for Implicit Function
> Theorem, it does not suffice to have the partial derivatives of first
> order continuous at the point, but it is needed that the continuity of
> those derivatives is satisfied in a neighborhood of the point. Has
> someone here came across with such an example?
>
> Thanks in advance,
>
> Paul

There is no such example, because the theorem (and the inverse function
theorem) does not need the partial derivatives to be continuous anywhere
except at the point. I don't know why most multivariable books throw in the
extra hypothesis.

Actually, so far as the inverse function theorem is concerned, it is only
for local injectivity (which can be proven by an easier more intuitive proof
than the standard one involving the fixed point theorem, as I recently
pointed out on this NG) that you I presume need the derivative to be
continuous at the point of interest. It is not too hard to show that you
don't need continuity of the derivative anywhere to show that the function
is open at the point (i.e., sends neighborhoods to neighborhoods, a sort of
local surjectivity condition). Indeed, intuitively it is clear that a small
sphere about the point has an image that wraps around the image of the
point, and as you deform the small sphere to the point, its image can't
deform to a point in the image of the function if there are points near the
image of the point which aren't in the image of the function. This is a very
pretty and simple algebraic topology application, and I don't understand why
it is so obscure (I could only find one reference to it on the Web after
searching for a good many minutes--it is definitely the obscure open mapping
theorem).

I don't even know of a function that has positive Jacobian determinant
everywhere on a neighborhood of a point that is not locally invertible at
the point. Is continuity of the derivative needed at all (provided the sign
of the Jacobian determinant stays the same)? I guess so, but I don't know.
(I asked June 6 for such a counterexample on this NG, and no one responded.)

Stephen A. Meigs

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