We have an additional condition that $f(x)>0$ except at $x=0$. $f$ vanishes at infinite order at x=0 means $f$ and its derivatives of any order equal 0 at x=0. Can you get $\sqrt{f}$ is smooth or not? Need a proof or counterexample.
If a smooth $f$ vanishes at infinite order at x=0, is $\sqrt{f}$ smooth?
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3Related:http://mathoverflow.net/questions/105438/square-root-of-a-positive-c-infty-function – 2017-01-24