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Function with range equal to whole reals on every open set

Hello,

My problem is the following "Is there a Real valued function with image of every open interval the whole real line?. Shall it exist, construct it"

I know so far that this equals finding uncountably many dense disjoint subset of R (in particular the preimages of the elements of R).

I shall be jolly thankful to anybody with a clue.

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The Conway base 13 function en.wikipedia.org/wiki/Conway_base_13_function – Andrey Rekalo Nov 14 2010 at 11:34
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 See also a previous MO question: mathoverflow.net/questions/32126/… – Andrey Rekalo Nov 14 2010 at 11:38
To find uncountably many disjoint dense subsets in $\mathbb R$, you may take uncountably many real numbers that are independent over $\mathbb Q$ and then translate $\mathbb Q$ by these numbers. – Bruno Martelli Nov 14 2010 at 11:42
This question is a duplicate of the question linked by Andrey. – Joel David Hamkins Nov 14 2010 at 11:46
See also this related MO question on whether the base 13 function is a measurable function mathoverflow.net/questions/32641/… – Joel David Hamkins Nov 14 2010 at 15:26

closed as exact duplicate by Andrey Rekalo, Joel David Hamkins, S. Carnahan♦ Nov 14 2010 at 12:05

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