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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 8, 2013 12:24 PM

On Sat, 07 Dec 2013 13:16:19 -0500, quasi <quasi@null.set> 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]. If f = 0 on P and f' = 0 on
Q then f = 0 on all of [0,1].

>
>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
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12/7/13 quasi
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12/7/13 quasi
12/8/13 David C. Ullrich
12/8/13 quasi
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12/9/13 David C. Ullrich
12/6/13 quasi
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12/9/13 quasi
12/9/13 quasi