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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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 quasi Posts: 12,067 Registered: 7/15/05
zero sets of infinitely differentiable functions
Posted: Dec 3, 2013 1:07 AM

I think the following is true ...

Proposition:

For any closed subset A of R there exists an infinitely
differentiable function f:R -> R such that A = f^(1)(0).

Proof:

If A = the empty set, let f(x) = 1 for all x, and we're done.

Thus, assume A is nonempty.

Let B = R\A. Then B is open so can be expressed as a countable
union of pairwise disjoint nonempty open intervals, say

B = B_1 U B_2 U ...

where the number of intervals in the union is either finite
or countably infinite.

Let B_n = (s_n,t_n) where s_n is either real or -oo, t_n is
either real or +oo, and s_n < t_n.

Let g_n: R -> R be an infinitely differentiable function such
that g_n = 0 on R\B_n (via a bump function construction).

Let d be the usual distance function on R.

For x in B_n, let d_n(x) = min(d(s_n,x),d(x,t_n)).

Define f:R -> R by

f(x) = 0 if x in A

f(x) = (d_n(x))*(g_n(x)) if x in B_n

Then f is infinitely differentiable and A = f^(-1)(0),
as required.

Is my proof correct?

If not, is the claim of the proposition true?

If so, here's a followup question ...

For an infinitely differentiable function f:R -> R, let f^(n)
denote the n'th derivative of f if n > 0 and f if n = 0.

Let A_0, A_1, A_2, ... be closed subsets of R such that for all
s,t in A_n with s < t, the set A_(n+1) has nonempty intersection
with the open interval (s,t).

Question:

Must there exist an infinitely differentiable function f:R -> R
such that, for all n, A_n = (f^(n))^(-1)(0)?

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