A few months ago I posted a question about analytic functions that I received from a bright high school student, which turned out to be studied and resolved by de Bruijn. Based on this positive resolution, I thought I might try my luck again and list three further questions that this student asked which do not seem to be trivially resolvable.

  1. Does there exist a smooth function {f: {\bf R} \rightarrow {\bf R}} which is nowhere analytic, but is such that the Taylor series {\sum_{n=0}^\infty \frac{f^{(n)}(x_0)}{n!} (x-x_0)^n} converges for every {x, x_0 \in {\bf R}}? (Of course, this series would not then converge to {f}, but instead to some analytic function {f_{x_0}(x)} for each {x_0}.) I have a vague feeling that perhaps the Baire category theorem should be able to resolve this question, but it seems to require a bit of effort. (Update: answered by Alexander Shaposhnikov in comments.)
  2. Is there a function {f: {\bf R} \rightarrow {\bf R}} which meets every polynomial {P: {\bf R} \rightarrow {\bf R}} to infinite order in the following sense: for every polynomial {P}, there exists {x_0} such that {f^{(n)}(x_0) = P^{(n)}(x_0)} for all {n=0,1,2,\dots}? Such a function would be rather pathological, perhaps resembling a space-filling curve. (Update: solved for smooth {f} by Aleksei Kulikov in comments. The situation currently remains unclear in the general case.)
  3. Is there a power series {\sum_{n=0}^\infty a_n x^n} that diverges everywhere (except at {x=0}), but which becomes pointwise convergent after dividing each of the monomials {a_n x^n} into pieces {a_n x^n = \sum_{j=1}^\infty a_{n,j} x^n} for some {a_{n,j}} summing absolutely to {a_n}, and then rearranging, i.e., there is some rearrangement {\sum_{m=1}^\infty a_{n_m, j_m} x^{n_m}} of {\sum_{n=0}^\infty \sum_{j=1}^\infty a_{n,j} x^n} that is pointwise convergent for every {x}? (Update: solved by Jacob Manaker in comments.)

Feel free to post answers or other thoughts on these questions in the comments.