Assume that $f\in C^{\infty}$ and that $M_n$ is a sequence such that $$\sum_{n=0}^{\infty}\frac{M_n}{(n+1)M_{n+1}}=\infty$$

and for certain compact neighborhood of the origin $U$ of $\mathbb{R}$, there is a constant $A$ such that for every $x\in U$, and $n\in\mathbb{N}$, \begin{equation}|g^{(n)}(x)|\leq n!A^nM_n,\end{equation}

where $g(x):=f(x)+f(2x)$. We say that $g$ is *quasianalytic* or that it belongs to the Denjoy-Carleman quasianalytic class $C_M(U)$.

Does it follow that $f\in C_M(V)$ for certain compact neighborhood of the origin $V$, i.e. $$|f^{(n)}(x)|\leq n!B^nM_n,$$ for certain constant $B$ and any $x\in V$, and $n\in\mathbb{N}$?

**Note:** For example, if $\forall n,\ M_n=1$ then the condition on $g$ implies it is analytic. Then it follows $f$ is also analytic and therefore $f\in C_M(V)$ for some $V$and some $B$.