
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

