A quasianalytic class with weakly smooth germs

In [O. Le Gal, J.-P. Rolin. An o-minimal structure which does not admit $C^{\mathrm{\infty }}$ cellular decomposition. In: Ann. Inst. Fourier 59 (2009), pp 543-562], the authors construct an o-minimal structure which does not admit smooth cell-decomposition. We explain in an appendix that their proof of o-minimality is incomplete and we give a complete proof in the main text.