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May 24 at 23:04 comment added Nate River Just read it! Great solution, and very natural idea despite the messiness.
May 24 at 23:03 vote accept Nate River
May 24 at 2:59 comment added Saúl RM Great. I just adapted the construction of the Cantor step function to a Cantor set of dimension 1 but measure $0$ (I already had an idea of how to construct that due to the question the other day). The proof turned out a bit ugly, but that seems difficult to avoid due to the intrinsic ugliness of decimal representations. It would be nice if there is a more elegant construction though
May 24 at 2:42 comment added Nate River Man I have absolutely no idea how you come up with these crazy constructions on the fly. I will read this and get back to you when I understand.
May 24 at 1:56 history answered Saúl RM CC BY-SA 4.0