Bump Functions and Partitions of Unity

The existence of a C^∞ partition of unity is one of the most important technical tools in the theory of C^∞ manifolds. It is the single feature that makes the behavior of C^∞ manifolds so different from real-analytic or complex manifolds. In this chapter we construct C^∞ bump functions on any manifold and prove the existence of a C∞ partition of unity on a compact manifold. The proof of the existence of a C^∞ partition of unity on a general manifold is more technical and is postponed to Appendix C.