On the definition of Schwartz functions

Question

$$ \mathcal{S}(\mathbb{R}^{n})=\Big\{f\in C^{\infty}(\mathbb{R}^{n})\,\Big|\; \sup_{\mathbb{R}^{n}}|x^{\alpha}D^{\beta}f(x)|<\infty\, \forall \alpha,\beta\, \text{multi indexes}\Big\}. $$ I can't find why is necessary the derivate condition; can you give me an example of a function such that $\sup_{\mathbb{R}^{n}}|x^{\alpha}f(x)| < \infty \,\forall \alpha \,\text{multi index}$ but exists $\beta$ such that $\sup_{\mathbb{R}^{n}}|x^{\alpha}D^{\beta}f(x)|=\infty$?

Answer 1

You can take $$ f \colon \mathbb{R} \to \mathbb{R} \, , \, f(x) = \cos\left(\mathrm{e}^{2 x^2}\right) \mathrm{e}^{-x^2} \, ,$$ and $\beta = 1$ .