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Let $f_n$ ($n=1,2,\dots$) be a sequence of functions $f_n\colon \mathbb R\to \mathbb R$ of class $C^1$ such that $f_n \rightrightarrows 0 $, $f_n' \rightrightarrows 0 $. Assume moreover that functions $f_n(\sqrt{x})$ ($n=1,2,...$) are also of class $C^1[0, \infty)$.

Is it then $[f_n(\sqrt{x})]' \rightrightarrows 0$ ?

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Try $f_{n}(x)=\frac{1}{n}\sin(x)$ to have a counterexample. – Lucien Sep 11 '12 at 9:31
But $f_n(\sqrt{x})$ is not of class $C^1$. – R.S Sep 11 '12 at 9:36
Sorry, I missread that. – Lucien Sep 11 '12 at 9:38
up vote 2 down vote accepted

Take $f_n(x)=a_n/(1+(x/b_n)^2)$ for suitable sequences of positive numbers $a_n,b_n$. To get $f_n\rightrightarrows0$ we need $a_n\to0$, for $f_n'\rightrightarrows0$ we need $a_n/b_n\to0$ (find the maximum of $|f_n'|$ to see it) and for $[f_n(\sqrt{x})]' \rightrightarrows 0$ we need $a_n/b_n^2\to0$. So a counterexample is $a_n=1/n$, $b_n=1/\sqrt{n}$.

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