TITLE: Uncountable preimage of every point 
QUESTION [8 upvotes]: Let $f:[0,1]\to [0,1]$ be a continuous function. Must it have a point $x$ that $f^{-1}(x)$ is at most countable?
Added: Must it have a point $x$ that $dim_H(f^{-1}(x))=0$ ? ($dim_H$ means the Hausdorff dimension)

REPLY [2 votes]: A simple example is to write any $x$ in $[0,1]$ as a binary sequence of $1$'s and $0$'s and then define continuous functions which takes odd (or even) values in the binary expansion of $x$.
That is, if $$x = a_1 a_2 a_3 a_4 a_5 \dots$$
then $$f(x) = a_1 a_3 a_5 \dots$$
This function is obviously continuous and has required property.
Also, the Peano curve could be defined in a similar fashion.