Composite Differentiable Functions

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.