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I'm studying the Gevrey class $G^s,\;s>1$, which is a subset of the $C^\infty$ class. I want to find an example of a function that is $C^\infty$ but not $G^s$.

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You can use Borel's theorem to find a function $g\in C^\infty(\mathbb{R})$ such that $$ g^{(j)}(0)=j!^{j}. $$ It's then clear this function can't be $G^s(\mathbb{R})$ for any $s$.

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  • $\begingroup$ Exactly! I'm familiar with Borel's theorem and this is a good example. Thanks. $\endgroup$
    – Jem Y
    Commented Dec 15, 2019 at 6:30
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    $\begingroup$ @JemY, if you find this a good example, then you should accept it. $\endgroup$
    – LSpice
    Commented Dec 15, 2019 at 12:41

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