Take the 2-minute tour ×
Mathematics Stack Exchange is a question and answer site for people studying math at any level and professionals in related fields. It's 100% free, no registration required.

Let $K\subset \mathbb{R}^n$ be a closed set, then is there existing a smooth function $f\in C^{\infty}(\mathbb{R}^n,\mathbb{R})$, such that $$ (1)\quad f\ge 0, $$ $$ (2) \quad f^{-1}(0)=K. $$

share|improve this question
    
Yes: first deal with the case where $K$ is the complement of an open ball, then jump to the general case writing $K$ as a countable union of such sets. –  Davide Giraudo Oct 12 '12 at 14:21
    
@DavideGiraudo: Did you mean countable intersection? –  Harald Hanche-Olsen Oct 12 '12 at 15:23
    
@HaraldHanche-Olsen Yes (now I can't edit the commnet anymore). –  Davide Giraudo Oct 12 '12 at 15:29
add comment

Your Answer

 
discard

By posting your answer, you agree to the privacy policy and terms of service.

Browse other questions tagged or ask your own question.