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Let $f \in C^\infty (\Bbb R) $, and $f(0) = 0$. Assume if $|w| \leq \delta$ then $| f(w) | \leq c |w|^a$ for some fixed $a \in \Bbb N$.

Now let $\| w \|_{L^\infty(\Bbb R)} \leq \delta$. Then can we derive an estimate for $\| f (w) \|_{C^k ( [0,\delta])}$ by using the inequality $|f(w)| \leq c |w|^a$ ?

Here $\| f (w) \|_{C^k ( [0,\delta])}$ means the supremum of all the functions up to differentiation order $k$ on $[0,\delta]$.

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    You seem to have a typo at the beginning of the second line.2012-12-18

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Inside the "envelope function" $ g(w)=\begin{cases}|w|^a,&\text{ if }|w|\leq1\\ |w|,&\text{ if }|w|>1\end{cases} $ you are free to make $f$ oscillate as much as you want, so $\|f\|_{C^k}$ can be made arbitrarily large.