On the non-extendability of quasianalytic germs

Let ${\mathcal{E}}_1(M{)}^{+}$ be the local ring of germs at 0 of functions belonging to a given Denjoy-Carleman quasianalytic class in a neighborhood of 0 in $[0,+\mathrm{\infty }[$. We show that the ring ${\mathcal{E}}_1(M{)}^{+}$ contains elements that cannot be extended quasianalytically in a neighborhood of 0 in $\mathbb{R}$, unless it coincides with the ring of real-analytic germs.