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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 9, 2013 10:28 AM

On Sun, 08 Dec 2013 15:11:22 -0500, quasi <quasi@null.set> wrote:

>dullrich wrote:
>>quasi wrote:
>>>dullrich wrote:
>>>>quasi wrote:
>>>>>quasi wrote:
>>>>>>dullrich wrote:
>>>>>>>quasi 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?

>>>>>>>
>>>>>>>[...]

>>>>
>>>>I suspect the answer is yes, but nothing springs to
>>>>mind for a proof.

>>>
>>>I have a proof in mind, but it's tricky.

>>
>>The answer is no. Let P = {0,1} and Q = [0,1].

>
>The problem requires both P,Q to be nowhere dense.

Aargh. Sorry...

>
>>If f = 0 on P and f' = 0 on Q then f = 0 on all of [0,1].
>
>Thus, it's not a counterexample.
>

>>>I'll outline it if requested.
>
>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