2026-07-072026-07-07http://salesiana.dossiersoluciones.com/handle/123456789/150045We 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 2Algebraic Geometry32B20, 58C27 (Primary) 32K15, 58C25 (Secondary)text