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Can we find a monotone function $f$ in $R^1$ satisfying that there is no continuous function $g$ equal to $f$ almost everywhere in every open interval $I$?

We know that a monotone function is differentiable almost everywhere,so I think the property of $f$ can not be too bad.but I don't know if $f$ can be approached by a continuous function for every open interval.I tried to construct some functions but didn't work...

Thank you in advance!

Maxwell
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  • I don't know what you mean by "a monotone function is derivative almost everywhere", but the Cantor function is continuous everywhere but nowhere continuous. – Bermudes Jan 16 '19 at 16:54
  • @Bermudes I think the OP means differentiable. See beginning of this question. – Dog_69 Jan 16 '19 at 16:58
  • @Bermudes thank you,but can these two condition hold at the same time?It seems like a contradiction...continuous and nowhere continuous... – Maxwell Jan 16 '19 at 17:16
  • @Dog_69 Yes.Thank you.I am not familiar with those words yet..sorry – Maxwell Jan 16 '19 at 17:19
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    @Maxwell I think he was wrong too. According to Wiki's page the Cantor function is continuous (uniformly continuous) but it is not completely continuous. – Dog_69 Jan 16 '19 at 17:48
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    Does a monotone increasing function which is discontinuous at all rational points do the job? The only possible discontinuities are jump discontinuities. I haven’t written a formal argument, though. – LinearOperator32 Jan 17 '19 at 06:09
  • @LinearOperator32 Thank you.I think you are right.I will have a try. – Maxwell Jan 17 '19 at 06:38

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