Let $f: R \rightarrow R$ be continuous on $R$ ( with $f(0)=0$ if necessary) and of class $C^\infty$ on $R \setminus \{0\}$. Assume that for each $n \in N$ the function $g_n(x)=x^n f(x)$, for $x \in R$, is of the class $C^n$ on $R$.
Is it true that $f$ is of the class $C^\infty$ on $R$ ?
Thanks