I want to know the example of cut-off function.
$\phi \in C^2 ([0,\infty))$ satisfies the followings :
(1) $\phi(x) =1$ on $[0,r]$
(2) $\phi(x) =0$ on $ x > 2r$
(3) $- C r^{-1} \phi^{1/2}(x) \leq \phi ' (x) \leq 0$ on $r \leq x \leq 2r $
(4) $| \phi '' (x) | \leq C r^{-2}$ on $r \leq x \leq 2r$
Is there an explicit example about cut-off function ?
Thank you in advance.