Assume we have two sets $V,W$, for which $V \subset \subset W \subset \subset \Omega \subset \mathbb{R}^n$ holds. Now we want to find a smooth function $\psi$, such that \begin{equation*} \begin{cases} \psi \equiv 1 \text{ in } V, \\ \psi \equiv 0 \text{ in } \mathbb{R}^n \setminus W, \\ \psi \in [0,1]. \end{cases} \end{equation*} holds.
How do we know that such a function exists? That problem occured to me in a proof in Chapter 6 of L. Evans' Book "Partial differential equations".