# A result attributed to Whitney

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?

• – Pietro Majer Feb 10 '15 at 19:00
• @Pietro: following your link, Peter Michor says:" Maybe, this result is hidden in Whitney, Hassler: Analytic extensions of differentiable functions defined in closed sets, Trans. AMS 36 (1934), 63--89." But where? – Nautilus Feb 10 '15 at 19:10
• I don't have Whitney's paper in front of me right now, but IIRC the extension operator constructed by Whitney when applied to the zero function (defined on a closed set $F$) has the property that the constructed function is comparable to the distance function from $F$. Do you need something else? – Willie Wong Feb 11 '15 at 3:13
• This is actually a very good elementary exercise. See Robert Israel's answer for $\mathbb R^n$, and glue by partition of unity: math.stackexchange.com/questions/791248/…. Whitney was concerned with much more subtle issues. – Igor Belegradek May 12 '15 at 0:19
• If you really like using Whitney's theorem, here is how (I will only do the case $M=\mathbb R^n$ here): Whitney proves that there is a function $f$ that vanishes on the given closed set $A$ and is analytic elsewhere. Then $f^2$ have the same property. Since zeros of analytic maps do not accumulate, they are isolated in the complement of $A$. Add a bump function to $f^2$ near every zero. This gives the desired function that vanishes precisely on $A$. – Igor Belegradek May 12 '15 at 0:47

## 2 Answers

Analytic extensions of differentiable functions defined in closed sets

see this discussion

and this related MSE question (first answer and comments)

• where in that paper is it found or how do you get it from the stated results? – Nautilus Feb 10 '15 at 15:39

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).