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Feb 25, 2020 at 15:58 comment added Gro-Tsen Furthermore, if you modify your construction just a little bit to make $g$ be a positive $C^\infty$ “bump” on each interval (connected component) of the complement of $C$, with all derivatives tending to $0$ at the edges of the interval, and with the height of the bumps tending fast enough to $0$, you can get $g$ hence $f$ to be both $C^\infty$, still with the same property.
Feb 25, 2020 at 12:55 history edited John Bentin CC BY-SA 4.0
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Feb 25, 2020 at 0:06 vote accept XT Chen
Feb 24, 2020 at 22:59 history edited Piotr Hajlasz CC BY-SA 4.0
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Feb 24, 2020 at 21:37 comment added Mateusz Kwaśnicki In case one needs a paper reference, virtually the same construction is carried out in Real Analysis Exchange 22(1) (1996-97): 404–405 by Javier Fernández de Bobadilla de Olazabal.
Feb 24, 2020 at 17:55 history edited Piotr Hajlasz CC BY-SA 4.0
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Feb 24, 2020 at 17:23 history edited Piotr Hajlasz CC BY-SA 4.0
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Feb 24, 2020 at 17:15 history undeleted Piotr Hajlasz
Feb 24, 2020 at 17:15 history edited Piotr Hajlasz CC BY-SA 4.0
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Feb 24, 2020 at 14:28 history deleted Piotr Hajlasz via Vote
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Feb 24, 2020 at 14:24 history deleted Piotr Hajlasz via Vote
Feb 24, 2020 at 14:24 history edited Piotr Hajlasz CC BY-SA 4.0
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Feb 24, 2020 at 14:15 history answered Piotr Hajlasz CC BY-SA 4.0