> **Possible Duplicate:**  
> [Function with range equal to whole reals on every open set](http://mathoverflow.net/questions/32126/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.