Bump function on set

Given an arbitrary compact set $K \subset \mathbb{R}^n$ and an open set $V \subset \mathbb{R}^n$, I want to construct a bump function $\zeta$ that is $1$ on $K$ and has its support in $V$, where I assume that $K \subset V$. Also it would be good to have $0 \le \zeta\le 1$ everywhere.

Could anybody give me a reference or explain to me how I can construct such a function?

• Evans PDEs and Lee introduction to smooth manifolds. – TKM Dec 12 '14 at 20:16
• @TKM on which page in Evans book? -Cannot find it – user159356 Dec 12 '14 at 20:23
• I think it's an exercise in Evans, Chapter 5. Though it may also be in the appendix, where he talks about convolutions. – Gyu Eun Lee Dec 12 '14 at 22:58
• @TobiasHurth : I think you mean $K\subset V$ instead of the other way around? – DisintegratingByParts Dec 13 '14 at 21:26

These bump functions, or test functions, are extremely important in distribution theory. They can be constructed using partitions of unity. I didn't find any good references online in this context, but if you can get hold of Hörmanders The Analysis of Linear Partial Differential Operators I'', then it's an excellent reference.