Take the 2-minute tour ×
MathOverflow is a question and answer site for professional mathematicians. It's 100% free, no registration required.

Possible Duplicate:
Function with range equal to whole reals on every open set


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.

share|improve this question

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

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.

The Conway base 13 function en.wikipedia.org/wiki/Conway_base_13_function –  Andrey Rekalo Nov 14 '10 at 11:34
See also a previous MO question: mathoverflow.net/questions/32126/… –  Andrey Rekalo Nov 14 '10 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 '10 at 11:42
This question is a duplicate of the question linked by Andrey. –  Joel David Hamkins Nov 14 '10 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 '10 at 15:26

Browse other questions tagged or ask your own question.