TITLE: "Converse" of Taylor's theorem
QUESTION [25 upvotes]: Let $f:(a,b)\to\mathbb{R}$. We are given $(k+1)$ continuous functions $a_0,a_1,\ldots,a_k:(a,b)\to\mathbb{R}$ such that for every $c\in(a,b)$ we can write $f(c+t)=\sum_{i=0}^k a_i(c)t^i+o(t^k)$ (for any $t$ in a neighbourhood of $0$). Can we conclude that $f$ is of class $C^k$?

REPLY [3 votes]: Another answer: You are prescribing a Whitney jet, and the Whitney extension theorem gives the answer:

Whitney, Hassler: Analytic extensions of differentiable functions defined in closed sets, Trans. AMS 36 (1934), 63--89.

Also, another reference building on the paper of Marcinkiewicz and Zygmund, is 

G. Glaeser: Etude de quelques Algebres Tayloriennes, J. Anal. Math. 11 (1958), 1-118.