I found this statement in a proof and thought it is clear, thinking of the absolute value function as an example, but I can not give a proof. Can someone give me a hint?
There is a function $f: R^6 \rightarrow R$ ($R$ are the real numbers) and $f\in C^2(R^6\setminus\left\{0\right\},R)$ and continous everywhere. Furthermore $f(0)=0$.
The claim is:
There is $n\in N$ such that $f^n\in C^2(R^6,R)$.
Note that
$$ f(x)=\frac{1}{\|x\|} \quad \text{if } x\neq 0, \quad f(0)=0 $$
belongs to $C^2(\mathbb{R}^6\setminus \{0\};\mathbb{R})$ but there is no $n\geq 1$ natural such that $f^n$ will be of class $C^2$ in a set including $0$.