Sur un problème d'O'Malley. (On a problem by O'Malley) (Q1088010)

A function \(f:<0,1>\to R\) has at a, a \(\in <0,1>\), an approximative maximum iff a is a point of the dispersion of the set \(\{t\in <0,1>:f(t)>f(a)\}.\) O'Malley asked whether each Darboux Baire one and almost continuous real function defined on \(<0,1>\) has an approximative maximum. The author gives a Darboux Baire one real function defined on \(<0,1>,\) which is almost continuous without any approximative maximum. There is also given a Darboux function with a countable set of discontinuity points having no local maximum.