Composite Differentiable Functions

Abstract

We introduce a new point of view towards Glaeser's theorem on composite $C^{\infty }$ functions [Ann. of Math. 1963], with respect to which we can formulate a ``$C^k$ composite function property" that is satisfied by all semiproper real analytic mappings. As a consequence, we see that a closed subanalytic set $X$ satisfies the $C^{\infty }$ composite function property if and only if the ring $C^{\infty }(X)$ of $C^{\infty }$ functions on $X$ is the intersection of all finite differentiability classes.

19 pages, hard copy available on request. amstex v 2