A note on extreme sets for quasi-continuous functions

Abtin Daghighi

doi:10.12988/pms.2014.4511

Abstract

It is known that for Ω a connected separable metric space, a function Ω → R which at every point attains a weak local extremum must have countable image. Let Ω ⊂ R n 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

Highlights

Behrends et al [1] gave a short proof of the fact that if Ω is a connected separable metric space, and if f : Ω → R attains weak local extremum at every point, f (Ω) is countable
In light of our examples we shall attempt to determine what can be said about the set of weak extrema given countability of the absolute value of a bounded quasi-continuous function, for arbitrary positive dimension n
The set A := j∈Æ Acj is open and it necessarily belongs to the set of weak local extrema of f

Summary

Introduction

Behrends et al [1] gave a short proof of the fact that if Ω is a connected separable metric space, and if f : Ω → R attains weak local extremum at every point, (the image) f (Ω) is countable. (ii) There exists an open set Up0 ⊂ Ω with p0 ∈ U p0, such that for any sequence {zj}j∈Æ in Up0 with zj → p0, it holds true that f (zj) → f (p0) It is not sufficient for a continuous real-valued function with weak local extrema (or even strict local maxima) which is dense, to reduce to a constant, see e.g. Posey & Vaughan [10] and Villani [11] who proved that for any countable subset of a metric space there is a real-valued continuous functions whose set of strict local maxima contains the given set. A special case would be to inquire sufficient conditions on the image of the absolute value in order for a quasi-continuous function to attain weak local extremum at every point (we shall later see, that such conditions must be quite restrictive and less interesting, which is why we will allow exceptional sets in our study). It is known that for any uncountable subset S ⊂ R there are only a countable set of points of S which are not accumulation points of S, any discrete set must be countable

Non-triviality of the problem

Result