A wide approximate continuity (Q1862666)

Denote by \(d_+(E,x)\), \(d_-(E,x)\) the right and left lower densities of a set \(E\subset {\mathbb R}\) at \(x\in{\mathbb R}\). Let \(f\) be a real-valued function defined on \(I=[0,1]\). For any \(x\in[0,1)\) let \[ u^+f(x)=\inf \{ r: d_+(\{ t: f(t)>r\},x)=0\},\;l^+f(x)=\sup\{ r: d^+(\{ t: f(t)<r\},x)=0\}. \] Similarly define \(u^-(f,x)\) and \(l^-(f,x)\) for \(x\in (0,1]\). A function \(f\) is called widely approximately continuous at \(x\) from the right (left) if \(u^+f(x)=l^+f(x)=f(x)\) (\(u^-f(x)=l^-f(x)=f(x)\)). \(f\) is approximately continuous at \(x\) if it is so from both sides at \(x\). \(f\) is widely approximate continuous on \(I\) if it is so at every point of \(I\). Properties of such functions are investigated in the paper. In particular, it is shown that such functions are \(DB_1\).