Bump Functions and Partitions of Unity

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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.