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Consider $f(r) = |r|^{m-1}r$ where $m \geq 1$.

Is it possible to find $C^\infty$ functions $f_n$, such that

- $f_n \to f$ uniformly on compact subsets of $\mathbb{R}$,
- $f_n’ \to f’$ uniformly on compact subsets of $\mathbb{R}$,
- $f_n'(0) > 0$,
- $f_n(0) = 0$

and do we get any upper/lower bounds on $f_n$ and $f_n’$, and if so, how do they depend on $n$?

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It is possible to find $f_n \in C^\infty$ such that 1,3,4 hold.

It is possible also to find $\tilde f_n \in C^\infty$ such that 1 and 2 hold.

But how about all four together?

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What about

$$

f_n(r) = \left(\sqrt{ r^2 + n^{-1}}\right)^{m-1}r

$$

By the mean value theorem, 2. 3. 4. together imply 1. So all you have to do is approximate $f’$ by smooth functions $g_n$ vanishing at $0$ (e.g. using convolution with symmetric test functions), then perturb very slightly $g_n$ so as to ensure $g_n(0)>0$ without ruining the convergence to $f’$, and finally set $f_n(t)=\int_0^t g_n(s) \,\mathrm{d}s$.

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