Lower semi-continuity

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Suppose LaTeX: x^k \rightarrow x. Of a function, LaTeX: f is lower semi-continuous if

LaTeX: \lim_{k \rightarrow \infty} \inf_{j \ge k} f(x^j) \ge f(x).

Of a point-to-set map, LaTeX: A, let LaTeX: N_{\varepsilon}[S] be a neighborhood of the set LaTeX: S: for each LaTeX: \varepsilon > 0, there exists LaTeX: K such that for all LaTeX: k \ge K, LaTeX: A(x) is a subset of LaTeX: N_{\varepsilon}[A(x^k)]. A pathology to show that the feasibility map may not be lower semi-continuous is:

LaTeX: A(b) = L_b(g) = \{x \in [-1, 1]: g(x) \le b\},

where LaTeX: g(x) = x if LaTeX: x is in LaTeX: [-1, 0], and LaTeX: g(x) = 0 if LaTeX: x \in [0,1].

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