I found this problem on an Italian forum and since then I struggled to solve it. The autor there claims it was proposed by Borel. At any rate the problem is as follows
(Borel?) For any $\{a_n\}_{n=0}^{\infty}\subseteq \mathbb R,\: x_0\in\mathbb R\,$ and $\,\varepsilon>0$ does there exist a $C^\infty$ function $f\colon\mathbb R\to\mathbb R$ such that $f^{(n)}(x_0)=a_n,\; \forall n\in\mathbb N\cup\{0\}\tag{1}$ and moreover $\left|f(x)-a_0\right|<\varepsilon,\;\forall x\in\mathbb R?$
Clearly even reference about problem are welcomed, but I strongly encourage anybody to think about it because really, when I met it for the first time, I thought: "this is a wonderful problem".
I do not have any clue towards the solution. I hope you will have fun with this.
Cheers.