Derivatives of flat functions

TL;DR

The paper investigates the properties of smooth functions on [0,1] that are flat at 0, showing such functions cannot have all derivatives positive on (0,1], and analyzing the zeroes of their derivatives.

Contribution

It establishes fundamental limitations on flat functions at zero, demonstrating the impossibility of having all derivatives positive and describing zeroes' behavior.

Findings

No smooth flat-at-zero function has all derivatives positive on (0,1].

Number of zeros of the n-th derivative grows to infinity as n increases.

Zeros of derivatives accumulate at zero as n approaches infinity.

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 $n \to \infty$.