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This question already has an answer here:

How can I prove that exists a monotonic non-decreasing function $f: [0,1] \rightarrow \mathbb R$ that isn't continuous in every rational of its domain?

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marked as duplicate by Gerry Myerson, Hagen von Eitzen, Davide Giraudo, Alex Becker, Namaste Feb 24 '13 at 22:46

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