Editing Semi-continuity (section)

=== Binary operations on semicontinuous functions ===
Let <math>f,g : X \to \overline{\R}</math>.
* If <math>f</math> and <math>g</math> are lower semicontinuous, then the sum <math>f+g</math> is lower semicontinuous<ref>{{cite book|last1=Puterman|first1=Martin L.|title=Markov Decision Processes Discrete Stochastic Dynamic Programming|url=https://archive.org/details/markovdecisionpr00pute_298|url-access=limited|date=2005|publisher=Wiley-Interscience|isbn=978-0-471-72782-8|pages=[https://archive.org/details/markovdecisionpr00pute_298/page/n618 602]}}</ref> (provided the sum is well-defined, i.e., <math>f(x)+g(x)</math> is not the [[indeterminate form]] <math>-\infty+\infty</math>). The same holds for upper semicontinuous functions.
* If <math>f</math> and <math>g</math> are lower semicontinuous and non-negative, then the product function <math>f g</math> is lower semicontinuous.  The corresponding result holds for upper semicontinuous functions.
* The function <math>f</math> is lower semicontinuous if and only if <math>-f</math> is upper semicontinuous.
* If <math>f</math> and <math>g</math> are upper semicontinuous and <math>f</math> is  [[Monotonic function|non-decreasing]], then the [[Function composition|composition]] <math>f \circ g</math> is upper semicontinuous. On the other hand, if <math>f</math> is not non-decreasing, then <math>f \circ g</math> may not be upper semicontinuous. For example take <math>f : \R \to \R </math> defined as <math>f(x)=-x</math>. Then <math>f </math> is continuous and  <math>f \circ g = -g</math>, which is not upper semicontinuous unless <math>g</math> is continuous.
* If <math>f</math> and <math>g</math> are lower semicontinuous, their (pointwise) maximum and minimum (defined by <math>x \mapsto \max\{f(x), g(x)\}</math> and <math>x \mapsto \min\{f(x), g(x)\}</math>) are also lower semicontinuous.  Consequently, the set of all lower semicontinuous functions from <math>X</math> to <math>\overline{\R}</math> (or to <math>\R</math>) forms a [[lattice (order)|lattice]].  The corresponding statements also hold for upper semicontinuous functions.