Take the 2-minute tour ×
Mathematics Stack Exchange is a question and answer site for people studying math at any level and professionals in related fields. It's 100% free, no registration required.

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

share|improve this question

1 Answer 1

up vote 3 down vote accepted

No, the absolute value function $x \mapsto \begin{cases} x, & \mbox{if } x \ge 0 \\ -x, & \mbox{if } x \lt 0 \end{cases}$ is a counterexample.

share|improve this answer
    
It's the first thing that comes to mind.. =) –  Patrick Da Silva Aug 11 '11 at 12:38

Your Answer

 
discard

By posting your answer, you agree to the privacy policy and terms of service.

Not the answer you're looking for? Browse other questions tagged or ask your own question.