# Lower semi-continuity

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 $LaTeX: x \in [0,1].$