Take the 2-minute tour ×
Mathematics Stack Exchange is a question and answer site for people studying math at any level and professionals in related fields. It's 100% free, no registration required.

Does there exist a continuous function $f : [0, 1] → [0, 1]$ such that the pre-image $f^{−1}(y)$ of any point $y \in [0, 1]$ is uncountable?

share|improve this question
2 related questions on MathOverflow: mathoverflow.net/questions/18666/…, mathoverflow.net/questions/47533/… –  Jonas Meyer Jan 19 '11 at 0:45
Sample paths of Brownian motion satisfy this property with probability one (if you cap and floor it to keep it in the range $[0,1]$ that is). –  George Lowther Jul 2 '11 at 11:12
add comment

1 Answer

up vote 15 down vote accepted


One nice way to see that is to take a Peano curve $c: [0,1] \to [0,1]^{2}$ (that is, a continuous surjection) and to compose it with the projection $p(x,y) = x$. Then $f = p \circ c$ will have the desired property.

Added. This is a folklore construction illustrating how far from the graphs we can actually draw (or imagine) a continuous function can be. As mentioned by Jonas in the comments this construction appears in at least two MO threads, namely here and here. I don't know where this example appeared first, I suspect that it can be found in Hausdorff's Mengenlehre, but Peano or Hilbert may have noticed it before that. They're not mentioning it in their original papers, though: Hilbert's paper and Peano's paper, links taken from the Wikipedia page on space-filling curves.

share|improve this answer
add comment

Your Answer


By posting your answer, you agree to the privacy policy and terms of service.

Not the answer you're looking for? Browse other questions tagged or ask your own question.