Editing Semi-continuity (section)

=== Optimization of semicontinuous functions ===

* The (pointwise) [[supremum]] of an arbitrary family <math>(f_i)_{i\in I}</math> of lower semicontinuous functions <math>f_i:X\to\overline{\R}</math> (defined by <math>f(x)=\sup\{f_i(x):i\in I\}</math>) is lower semicontinuous.<ref>{{cite web |title=To show that the supremum of any collection of lower semicontinuous functions is lower semicontinuous |url=https://math.stackexchange.com/q/1662726}}</ref>

:In particular, the limit of a [[monotone increasing]] sequence <math>f_1\le f_2\le f_3\le\cdots</math> of continuous functions is lower semicontinuous.  (The Theorem of Baire below provides a partial converse.)  The limit function will only be lower semicontinuous in general, not continuous.  An example is given by the functions <math>f_n(x)=1-(1-x)^n</math> defined for <math>x\in[0,1]</math> for <math>n=1,2,\ldots.</math>

:Likewise, the [[infimum]] of an arbitrary family of upper semicontinuous functions is upper semicontinuous.  And the limit of a [[monotone decreasing]] sequence of continuous functions is upper semicontinuous.

* If <math>C</math> is a [[compact space]] (for instance a closed bounded interval <math>[a, b]</math>) and <math>f : C \to \overline{\R}</math> is upper semicontinuous, then <math>f</math> attains a maximum on <math>C.</math> If <math>f</math> is lower semicontinuous on <math>C,</math> it attains a minimum on <math>C.</math>
:(''Proof for the upper semicontinuous case'': By condition (5) in the definition, <math>f</math> is continuous when <math>\overline{\R}</math> is given the left order topology.  So its image <math>f(C)</math> is compact in that topology.  And the compact sets in that topology are exactly the sets with a maximum.  For an alternative proof, see the article on the [[extreme value theorem]].)