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Given that the space of indefinitely differentiable functions with compact support $C^\infty_c([0,1])$ is dense in $L^1([0,1])$, how can I find a sequence in $C^\infty_c([0,1])$ that converges to the integrable function $f(x)=1/\sqrt{x}$ if $x\in]0,1]$, $f(0)=0$?. The usual truncation method does not work because it produces non-$C^\infty$ functions.

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Let $\phi_n:[0,1]\to[0,1]$ be a sequence of bump functions that is in $C_c^\infty([0,1])$ such that $\phi_n(x) = 1$ if $\frac1n<x<1-\frac1n$, and $\phi_n(x) = 0$ if $x < \frac1{2n}$ or $x>1-\frac1{2n}$.

Then consider $f_n = \phi_n f$.

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  • $\begingroup$ Of course! Truncation with bump functions... I should have get this by myself. Thank you very much! $\endgroup$ – Grimolatto Dec 28 '13 at 6:27

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