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