SAC property and approximate semicontinuity. (English) Zbl 1013.26005

Prus-Wiśniowski, Franciszek; Szkibiel, Grzegorz

Real Anal. Exch. 26(2000-2001), No. 2, 947-962 (2001).

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 \(x_0\).

Reviewer: Zbigniew Grande (Bydgoszcz)

MSC:

26A15

Continuity and related questions (modulus of continuity, semicontinuity, discontinuities, etc.) for real functions in one variable

Keywords:

approximate semicontinuity; approximate continuity