Beschreibung:
<jats:title>Abstract</jats:title><jats:p>We prove that if f : ℝ<jats:sup><jats:italic>N</jats:italic></jats:sup> → ℝ̄ is quasiconvex and <jats:italic>U</jats:italic> ⊂ ℝ<jats:sup><jats:italic>N</jats:italic></jats:sup> is open in the density topology, then sup<jats:sub><jats:italic>U</jats:italic></jats:sub> ƒ = ess supU f ; while inf<jats:sub><jats:italic>U</jats:italic></jats:sub> ƒ = ess sup<jats:sub><jats:italic>U</jats:italic></jats:sub> ƒ if and only if the equality holds when U = RN: The first (second) property is typical of lsc (usc) functions, and, even when U is an ordinary open subset, there seems to be no record that they both hold for all quasiconvex functions.</jats:p><jats:p>This property ensures that the pointwise extrema of f on any nonempty density open subset can be arbitrarily closely approximated by values of ƒ achieved on “large” subsets, which may be of relevance in a variety of situations. To support this claim, we use it to characterize the common points of continuity, or approximate continuity, of two quasiconvex functions that coincide away from a set of measure zero.</jats:p>