A Note on Extreme Sets for Quasi-Continuous Functions

It is known that for Ω a connected separable metric space, a function Ω → R which at every point attains a weak local extremummust have countable image. Let Ω ⊂ R be a bounded domain and f : Ω → C a measurable, bounded, quasi-continuous function. For n ∈ Z+, we show that if |f | has countable image then f attains weak local extremum at each point of a dense open subset. We show by examples the optimality of the result, in the sense that strengthening the countability condition to discreteness (or even finiteness) will not affect the dense open condition. Conversely it is easily verified that if f attains weak local extremum off a countable set then |f | has countable image. Mathematics Subject Classification: 26A15, 54C08, 54C30