# A question about smoothness

$f$ is a smooth function on $[0,+\infty)$ and $f(x)>0$ for all $x>0$. Then does the following equivalence hold :

$\phi(x,y)=f(\sqrt{x^2+y^2})$ is smooth if and only if $f^{(k)}(0)=0$ for all odd number $k$.

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The answer is yes. The "if" direction is e.g. Lemma 1.7.3 in the book "The Geometry of Total Curvature on Complete Open Surfaces" which can be found in google books. The "only if" direction follows by induction essentially because $\sqrt{x^2+y^2}$ is nonsmooth at the origin. Neither direction is hard; it is a good calculus exercise. – Igor Belegradek Oct 10 '12 at 23:16
Sorry, I meant Lemma 7.1.3. – Igor Belegradek Oct 10 '12 at 23:18