Smooth cutoff function

Let $A\subseteq U\subseteq\mathbb R^n$ , with $U$ open. A smooth cutoff for $A$ supported in $U$ is a smooth function $\chi:\mathbb R^n\to[0,1]$ that equals $1$ on a neighborhood of $A$ and satisfies $\operatorname{supp}\chi\subseteq U$ .

Compact and noncompact versions

When $A$ is compact, one can choose $\chi$ to have compact support in $U$ . More generally, cutoffs adapted to unbounded regions need not have compact support. The support and plateau requirements must therefore be specified separately.

One-dimensional transition

Let $h(t)=e^{-1/t}$ for $t>0$ and $h(t)=0$ for $t\le0$ . Then

\vartheta(t)=\frac{h(t)}{h(t)+h(1-t)}

is smooth, equals $0$ for $t\le0$ , and equals $1$ for $t\ge1$ . The denominator is positive everywhere; smoothness of $h$ at zero follows from the flat-exponential construction. Translating and rescaling transitions produces plateaus with prescribed margins.

References

Richard Schwartz, Partitions of Unity (Brown lecture notes).