Editing Semi-continuity (section)

=== Inner and outer semicontinuity ===

A set-valued function <math>F: \mathbb{R}^m \rightrightarrows \mathbb{R}^n</math> is called ''inner semicontinuous'' at <math>x</math> if for every <math>y \in F(x)</math> and every convergent sequence <math>(x_i)</math> in <math>\mathbb{R}^m</math> such that <math>x_i \to x</math>, there exists  
a sequence <math>(y_i)</math> in <math>\mathbb{R}^n</math> such that <math>y_i \to y</math> and <math>y_i \in F\left(x_i\right)</math> for all sufficiently large <math>i \in \mathbb{N}.</math><ref name="goebelSetvalued"/><ref group="note">In particular, there exists <math>i_0 \geq 0</math> such that <math>y_i \in F(x_i)</math> for every natural number <math>i \geq i_0,</math>. The  necessisty of only considering the tail of <math>y_i</math> comes from the fact that for small values of <math>i,</math> the set <math>F(x_i)</math> may be empty.</ref>

A set-valued function <math>F: \mathbb{R}^m \rightrightarrows \mathbb{R}^n</math> is called ''outer semicontinuous'' at <math>x</math> if for every convergence sequence <math>(x_i)</math> in <math>\mathbb{R}^m</math> such that <math>x_i \to x</math> and every convergent sequence <math>(y_i)</math> in <math>\mathbb{R}^n</math> such that <math>y_i \in F(x_i)</math> for each <math>i\in\mathbb{N},</math> the sequence <math>(y_i)</math> converges to a point in <math>F(x)</math> (that is, <math>\lim _{i \to \infty} y_i \in F(x)</math>).<ref name="goebelSetvalued"/>
<!--The definitions of upper and lower semicontinuity are defined using open neighborhoods, where as inner and outer semicontinuity are defined using convergent sequences.-->