TITLE: A result attributed to Whitney
QUESTION [10 upvotes]: One of the basic results of real analysis says that any closed subset of a smooth ($C^\infty$) manifold $M$ is the set of zeros of some map $\lambda\in C^\infty(M;[0,1])$. This result (or some equivalent variations of it) is usually attributed to Hassler Whitney.
Question: exactly where (title of Whitney's paper and page) is this stated?

REPLY [2 votes]: According to J.-C. Tougeron, page 73 (Théorème 2.2) of Idéaux de fonctions différentiables, Ergebnisse, Band 71, Whitney's paper is

Analytic extensions of differentiable functions defined in closed sets. Trans. Amer. Math. Soc. 36 (1934), pp 63-89.

Actually, Whitney's Theorem is stronger in two ways. On the one hand, every $C^m$-function $F:K\rightarrow{\mathbb R}$ ($K$ the closed set) can be matched at order $m$ by a $C^m$-function $W(F):M\rightarrow{\mathbb R}$ which is $C^\infty$ away from $K$. On the other hand, the operator $W$ can be chosen linear and continuous (for the $C^m$-norms).