Upper semi-continuity

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(or upper semi-continuous, abbr. usc). Suppose LaTeX: \textstyle \{x^k\} \to x. Of a function, LaTeX: \textstyle \limsup f(x^k) \le f(x). Of a point-to-set map, let LaTeX: \textstyle \mbox{N}_e\lbrack\mbox{S}\rbrack be a neighborhood of the set S. For each e > 0, there exists K such that for all LaTeX: \textstyle k > \mbox{K}, \mbox{A}(x^k) is a subset of LaTeX: \textstyle \mbox{N}_e\lbrack\mbox{A}(x)\rbrack. Here is an example of what can go wrong. Consider the feasibility map with

g(x) =
<pre>(x+\sqrt{2})^2 - 1, & \mbox{if } x < 0 \\
e^{-x}, & \mbox{if } x \ge 0

Note g is continuous and its level set is LaTeX: \textstyle \lbrack -\sqrt{2} - 1, -\sqrt{2} + 1 \rbrack. However, for any

LaTeX: \textstyle b > 0, \{x: g(x) \le b\} = \lbrack - \sqrt{2} - 1 - b, - \sqrt{2} + 1 + b \rbrack \lor \lbrack - \log b, \inf),

which is not bounded. The map fails to be usc at 0 due to the lack of stability of its feasibility region when perturbing its right-hand side (from above).

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