Bump function - Knowlpedia

Let $M$ be a smooth manifold. A bump function is a smooth map $\varphi:M\to\mathbb R$ whose support

\operatorname{supp}\varphi=\overline{\{p\in M:\varphi(p)\ne0\}}

is a compact set, where the bar denotes closure. In localized constructions one usually requires more: for a closed set $A\subseteq M$ and an open set $U\supseteq A$ , a bump function for $A$ supported in $U$ satisfies $0\le\varphi\le1$ , equals $1$ on a neighborhood of $A$ , and has $\operatorname{supp}\varphi\subseteq U$ .

Existence on smooth manifolds

If $A\subseteq U\subseteq M$ , with $A$ closed and $U$ open, and $A$ is compact, then such a function exists. The proof first constructs Euclidean cutoff functions and then combines local constructions using a partition of unity.

Uses and constructions

Bump functions extend locally defined objects, splice constructions without changing them on a chosen region, and build partitions of unity. If $s$ is a local section over $U$ and $\varphi$ has support contained in $U$ , then $\varphi s$ , extended by zero, is a global smooth section.

Conventions and non-examples

Some authors use “cutoff function” for the more specific $0$ -to- $1$ function adapted to $A\subseteq U$ , and others do not require every cutoff to have compact support. The constant function $1$ is therefore a bump function precisely when $M$ is compact. A compactly supported continuous function with a corner is not a smooth bump function.

References

John M. Lee, Introduction to Smooth Manifolds, 2nd ed., Springer, 2012. DOI record. Relevant: smooth functions, bump functions, and partitions of unity.
Jet Nestruev, Smooth Manifolds and Observables, Springer, 2003. DOI record. Relevant: “Cutoff and Other Special Smooth Functions.”

Euclidean cutoff conventions

A smooth cutoff specifies a plateau and a surrounding support region; compact support is imposed separately when needed. Flat exponentials give explicit Euclidean examples.