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We prove two main results on Denjoy–Carleman classes: (1) a composite function theorem which asserts that a function
in a quasianalytic Denjoy–Carleman class
, which is formally composite with a generically submersive mapping
, at a single given point in the source (or in the target) of
can be written locally as
belongs to a shifted Denjoy–Carleman class
; (2) a statement on a similar loss of regularity for functions definable in the
-minimal structure given by expansion of the real field by restricted functions of quasianalytic class
. Both results depend on an estimate for the regularity of a
of the equation
as above. The composite function result depends also on a quasianalytic continuation theorem, which shows that the formal assumption at a given point in (1) propagates to a formal composition condition at every point in a neighbourhood.
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