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A counterexample on the semicontinuity of minima

Authors: Fernando Luque-Vásquez and Onésimo Hernández-Lerma
Journal: Proc. Amer. Math. Soc. 123 (1995), 3175-3176
MSC: Primary 49J45; Secondary 49K40
MathSciNet review: 1301515
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Abstract: Let X and Y be metric spaces, $ \Phi $ a multifunction from X to Y, and v a real-valued function on $ X \times Y$. We give an example in which $ \Phi $ is continuous, and v is continuous, inf-compact and bounded below, but the minimum function $ {v^ \ast }(x): = {\inf _{y \in \Phi (x)}}v(x,y)$ on X is not lower semicontinuous.

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Keywords: Minimization problem, multifunctions, measurable selections
Article copyright: © Copyright 1995 American Mathematical Society