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We prove two main results on Denjoy–Carleman classes: (1) a composite function theorem which asserts that a function 
$f(x)$ in a quasianalytic Denjoy–Carleman class 
${\mathcal{Q}}_{M}$, which is formally composite with a generically submersive mapping 
$y=\unicode[STIX]{x1D711}(x)$ of class 
${\mathcal{Q}}_{M}$, at a single given point in the source (or in the target) of 
$\unicode[STIX]{x1D711}$ can be written locally as 
$f=g\circ \unicode[STIX]{x1D711}$, where 
$g(y)$ belongs to a shifted Denjoy–Carleman class 
${\mathcal{Q}}_{M^{(p)}}$; (2) a statement on a similar loss of regularity for functions definable in the 
$o$-minimal structure given by expansion of the real field by restricted functions of quasianalytic class 
${\mathcal{Q}}_{M}$. Both results depend on an estimate for the regularity of a 
${\mathcal{C}}^{\infty }$ solution 
$g$ of the equation 
$f=g\circ \unicode[STIX]{x1D711}$, with 
$f$ and 
$\unicode[STIX]{x1D711}$ 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.
