TITLE: Continuity of real functions
QUESTION [5 upvotes]: The following question concerns that without $ZF+DC$, can every function be  "a little bit" continuous?

Question Is it consistent with $ZF+DC$ that for any function $f:[0,1]\to [0,1]$ and any positive measure set $A$, there is a positive measure closed set $B\subseteq A$ so that $f^{-1}(B)$ is closed?

How about in Solovay's model?

REPLY [7 votes]: Any Borel $f:[0, 1] \to [0, 1]$ satisfying $f^{-1}[\{y\}]$ is dense in $[0, 1]$ for every $y \in [0, 1]$ is a counterexample. For example, $f(x) = \limsup_n \frac{x_1 + x_2 + \dots + x_n}{n}$ where $x = 0.x_1 x_2 \dots$ is the binary expansion of $x$.