Is there a Real valued function with image of every open interval the whole real line [duplicate]

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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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asked Nov 14, 2010 at 11:29

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$\begingroup$ The Conway base 13 function en.wikipedia.org/wiki/Conway_base_13_function $\endgroup$
– Andrey Rekalo
Commented Nov 14, 2010 at 11:34
1

$\begingroup$ See also a previous MO question: mathoverflow.net/questions/32126/… $\endgroup$
– Andrey Rekalo
Commented Nov 14, 2010 at 11:38
$\begingroup$ 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. $\endgroup$
– Bruno Martelli
Commented Nov 14, 2010 at 11:42
$\begingroup$ This question is a duplicate of the question linked by Andrey. $\endgroup$
– Joel David Hamkins
Commented Nov 14, 2010 at 11:46
$\begingroup$ See also this related MO question on whether the base 13 function is a measurable function mathoverflow.net/questions/32641/… $\endgroup$
– Joel David Hamkins
Commented Nov 14, 2010 at 15:26

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