On Wed, 04 Dec 2013 06:36:23 -0500, quasi <email@example.com> 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.
Which doesn't quite answer your question, but it's 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