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arXiv · 1805.01879

Derivatives of flat functions

Hiroki Kodama·Kazuo Masuda·Yoshihiko Mitsumatsu
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Abstract

We remark that there is no smooth function $f(x)$ on $[0, 1]$ which is flat at $0$ such that the derivative $f^{(n)}$ of any order $n\geq 0$ is positive on $(0,1]$. Moreover, the number of zeros of the $n$-th derivative $f^{(n)}$ grows to the infinity and the zeros accumulate to $0$ when $n \to \infty$.

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Hiroki Kodama, Kazuo Masuda, Yoshihiko Mitsumatsu. 2018-05-04. Derivatives of flat functions. https://arxiv.org/abs/1805.01879

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