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I created this question, but, I do not know the answer:

Is there a function $f: \mathbb R \to \mathbb R$ such that for every interval $I \subseteq \mathbb R$ we have $f(I)=\mathbb R$?

It seems to me that existence of such a function would violate almost everything that I know about analysis, but, on the other hand, it would be awesome if there is at least one such $f$, because it is known that there are many "pathological" examples in analysis. Also, many here are skilled in analysis more than I am, so I can expect an answer, curious about what it will turn to be.

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marked as duplicate by Kabo Murphy real-analysis Jun 18 at 5:35

This question has been asked before and already has an answer. If those answers do not fully address your question, please ask a new question.

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There's for instance Conway's base-$13$ function.

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  • $\begingroup$ Are there any other examples? Or how to obtain them? I thought that it will be very hard to find such a function. $\endgroup$ – Grešnik Jun 18 at 3:24

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