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1) Is there sequences of continuous function on $[0,1]$ that converge converge to a function that is nowhere continuous ?

2) Same question on all $\mathbb R$.

I was looking for a sequence that converge to $1_{\mathbb Q\cap [0,1]}$, but not conclusive.

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    $\begingroup$ Do you know Baire's theorem? $\endgroup$ – Daniel Fischer Feb 26 '17 at 16:39
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As you can see in this discussion, the number of discontinuous points can be dense in $[0,1]$, but the set of discontinuous points must be a meagre set by the Baire-Osgood theorem. As a subset of a meagre or a countable union of meagre sets are also meagre (thank you wikipedia), I suppose $\Bbb R$ and $[0,1]$ are not meagre, otherwise this concept would be useless. So no, it is not possible.

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