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Topic: zero sets of infinitely differentiable functions
Replies: 31   Last Post: Dec 9, 2013 4:14 PM

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 David C. Ullrich Posts: 3,555 Registered: 12/13/04
Re: zero sets of infinitely differentiable functions
Posted: Dec 6, 2013 12:05 PM

On Wed, 04 Dec 2013 06:36:23 -0500, quasi <quasi@null.set> wrote:

>Let's try a special case ...
>
>Question:
>
>If P,Q are closed, nowhere dense subsets of R such that between
>any two distinct elements of P there is an element of Q, must
>there exist a differentiable function f:R -> R such that
>f^(-1)(0) = P and (f')^(-1)(0) = Q?

Surely yes. In fact I'd be willing to bet that this follows
easily from what's been proved already, by something
sort of like the following:

Start with g such that g = 0 on P and g > 0 on R\P.
Now if h tends to infinity fast enough at +- infinty
then the function hg satisfies hg = 0 on P, hg > 0
on R\P, and hg > 1 on Q.

So if you choose phi appropriately then the
function psi = phi(gh) satisfies psi = 0 on P,
psi > 0 on R\P, 0 <= psi <= 1 everywhere,
and psi = 1 on Q.

an example of the sort of jiggling with the previous
result I have in mind... go for it.

>
>Remark:
>
>If the answer is "no", then the answer to my previously posted
>more general question is also "no".
>
>quasi

Date Subject Author
12/3/13 quasi
12/3/13 Virgil
12/3/13 quasi
12/3/13 quasi
12/3/13 quasi
12/3/13 quasi
12/3/13 quasi
12/9/13 quasi
12/3/13 David C. Ullrich
12/3/13 quasi
12/4/13 quasi
12/6/13 David C. Ullrich
12/6/13 quasi
12/6/13 quasi
12/6/13 quasi
12/7/13 quasi
12/7/13 David C. Ullrich
12/7/13 quasi
12/8/13 David C. Ullrich
12/8/13 quasi
12/8/13 quasi
12/8/13 quasi
12/8/13 quasi
12/8/13 quasi
12/8/13 quasi
12/8/13 quasi
12/9/13 quasi
12/9/13 David C. Ullrich
12/6/13 quasi
12/6/13 David C. Ullrich
12/9/13 quasi
12/9/13 quasi