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

Possible Duplicate:
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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## marked as duplicate by Andrey Rekalo, Joel David Hamkins, S. Carnahan♦Nov 14 '10 at 12:05

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 '10 at 11:42