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.

Let ${{C}^{M}}$ denote a Denjoy–Carleman class of ${{C}^{\infty }}$ functions (for a given logarithmically-convex sequence $M\,=\,\left( {{M}_{n}} \right))$. We construct: (1) a function in ${{C}^{M}}\left( \left( -1,\,1 \right) \right)$ that is nowhere in any smaller class; (2) a function on $\mathbb{R}$ that is formally ${{C}^{M}}$ at every point, but not in ${{C}^{M}}\left( \mathbb{R} \right)$; (3) (under the assumption of quasianalyticity) a smooth function on ${{\mathbb{R}}^{p}}\,\left( p\,\ge \,2 \right)$ that is ${{C}^{M}}$ on every ${{C}^{M}}$ curve, but not in ${{C}^{M}}\left( {{\mathbb{R}}^{p}} \right)$.