The following question concerns that without $ZF+DC$, can every function be "a little bit" continuous?



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> **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?
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How about in Solovay's model?



Continuity of real functions
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 $x$.


