# Existence of a continuous function with pre-image of each point uncountable

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?

-
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
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.