Let $f$ be a non-negative infinitely smooth function on the real line. Is it true that for any constant $\alpha$ the function $f^\alpha$ is infinitely smooth?
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$\begingroup$ No. it is not true. $\endgroup$– Alex DegtyarevCommented Aug 9, 2015 at 17:27
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2 Answers
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No. Let $f(x)=x^2,$ while $a=1/4.$
answered Aug 9, 2015 at 13:16
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There is an interesting variant of the question: Suppose that the funktion is smooth, positive, and flat at all zeroes (i.e. all derivatives vanish). Is the square root again smooth?
Glaeser proved that it is continuously differebtiable and gave an example wehre it is not twice differentiable.
answered Aug 9, 2015 at 18:48
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$\begingroup$ There is a big theory of this sort of thing, going back to Whitney, I believe. $\endgroup$ Commented Aug 9, 2015 at 19:08