On real analytic functions on closed subanalytic domains.

We show that a function defined on a closed uniformly polynomially cuspidal set in is real analytic if and only if is smooth and all its composites with germs of polynomial curves in are real analytic. The degree of the polynomial curves needed for this is effectively related to the regularity of the boundary of . For instance, if the boundary of is locally Lipschitz, then polynomial curves of degree 2 suffice.

Nonlinear Conditions for Ultradifferentiability.

A remarkable theorem of Joris states that a function is if two relatively prime powers of are . Recently, Thilliez showed that an analogous theorem holds in Denjoy-Carleman classes of Roumieu type. We prove that a division property, equivalent to Joris's result, is valid in a wide variety of ultradifferentiable classes.