Let $f\in L^p(\mathbb{R})$. I know I cannot find a sequence of $C^\infty$ functions that converges uniformly to $f$ in general. I also know that I can find a sequence of sequence of $C^\infty$ functions that converge to $f$ in $L^p$. I know that I can find a subsequence of the latter that converges a.e. to $f$. But what about a convergence everywhere? What would be an example of a function where I cannot have convergence in every point by a sequence of $C^{\infty}$-functions? Maybe $\mathbb{1}_{\mathbb{Q}}$ (characteristic function on the rationals)? How could I prove this (the result with $\mathbb{1}_\mathbb{Q}$)? Is there something more simple?
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This has been studied by Baire and has nothing to do with $L^p$ spaces. Search for "Baire one functions". If you read French, as your username suggests, you will have access to a lot more material. – Giuseppe Negro Oct 23 '20 at 10:44
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Thank you. I will look at the concept – roi_saumon Oct 23 '20 at 14:14
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This answer could be useful. – Giuseppe Negro Dec 19 '20 at 11:14