A remarkable theorem of Joris states that a function $f : \mathbb R^d \to \mathbb R$ is of class $C^\infty$ if two coprime powers of $f$, e.g., $f^2$ and $f^3$, are of class $C^\infty$. Naively dividing $f^3$ by $f^2$ clearly does not prove this result, but there is a way to make this strategy work. This path leads through the complex domain by almost analytic extension and holomorphic approximation. I will explain this approach and show that it can similarly be applied to a wide

variety of smooth regularity classes $\mathcal C$, including for instance quasianalytic Denjoy--Carleman classes. Furthermore, I will present a full characterization of the analytic germs $\Phi : (\mathbb R,0) \to (\mathbb R^n,0)$ with the property that $\Phi \circ f \in \mathcal C$ implies $f \in \mathcal C$, for all continuous function germs $f$, in terms of a condition on the support of

the Taylor series of $\Phi$.