Does every Borel function have a somewhere continuous modification! (Q1803986)

A Borel function \(f: \mathbb{R}\to \mathbb{R}\) has property (Ber) if, for each set \(A\subset \mathbb{R}\) with non-zero Lebesgue measure and each nonempty open set \(B\subset \mathbb{R}\), the set \(f^{-1}(A)\cap B\) has non-zero Lebesgue measure. Let \((\text{Ber}^*)\) be the analogue of property (Ber) for Borel functions from \([0,1[\) into \(\mathbb{R}\). In this paper the authors construct a function \(f\) with property \((\text{Ber}^*)\) (the periodic continuation of such a function to \(\mathbb{R}\) will have property (Ber)).