SAC property and approximate semicontinuity (Q5917425)

For a bounded measurable function \(f:{\mathcal R} \to {\mathcal R}\) and for real \(r > 0\) let \[ p_r(x) = \sup \left\{ s\in (0,1]:\biggl|\frac{1}{h}\int_x^{x+h}f(t) dt- f(x)\biggr|< r\;\text{for }0 < |h|< s\right\} . \] If in the above formula we write \(\leq r\) then we define \(q_r(x)\). It is proved that if \(f\) is approximately continuous at \(x_0\) then \(p_r\) is approximately lower semicontinuous at \(x_0\) and \(q_r\) is approximately upper semicontinuous at \(x_0\).