It follows from Whitney extension theorem that for every closed set $ C \subseteq \mathbb{R}^n $ and for every $ k \geq 1 $ there exists a function $ f \in C^k(\mathbb{R}^n) $ such that $ C = \{x : f(x)=0 \} $ and $ D^if(x) =0 $ for every $ x \in C $ and $ i = 1, \ldots , k $.


Is it possible to replace $ k $ with $ \infty $ in the statement above?



Smooth functions with zeros of infinite order on a closed set
Yes, for example, take an open cover of the complement of $C$
by countably many open balls of radius $r\_i$ centered at $v\_i$, and use a partition of unity $\{f\_i\}\_{i∈I}$ subordinate to this cover to glue bump functions
$\exp(-(\max(r\_i^2-‖x-v\_i‖^2,0))^{-2})$ into a globally smooth function
$$∑\_{i∈I}f\_i \exp(-(\max(r\_i^2-‖x-v\_i‖^2,0))^{-2}).$$


