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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?

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2 related questions on MathOverflow: mathoverflow.net/questions/18666/…, mathoverflow.net/questions/47533/… –  Jonas Meyer Jan 19 '11 at 0:45
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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

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up vote 15 down vote accepted

Yes.

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.

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