  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.
