Editing Semi-continuity (section)

=== Other properties ===

* ('''Theorem of Baire''')<ref group="note">The result was proved by René Baire in 1904 for real-valued function defined on <math>\R</math>.  It was extended to metric spaces by [[Hans Hahn (mathematician)|Hans Hahn]] in 1917, and [[Hing Tong]] showed in 1952 that the most general class of spaces where the theorem holds is the class of [[perfectly normal space]]s.  (See Engelking, Exercise 1.7.15(c), p. 62 for details and specific references.)</ref> Let <math>X</math> be a [[metric space]].  Every lower semicontinuous function <math>f:X\to\overline{\R}</math> is the limit of a point-wise  [[monotone increasing|increasing]] sequence of extended real-valued continuous functions on <math>X.</math> In particular, there exists a sequence <math>\{f_i\}</math> of continuous functions <math>f_i : X \to \overline\R</math> such that 
:<math display="block">f_i(x) \leq f_{i+1}(x) \quad \forall x \in X,\ \forall i = 0, 1, 2, \dots</math> and 
:<math display="block">\lim_{i \to \infty} f_i(x) = f(x) \quad \forall x \in X. </math>
:If <math>f</math> does not take the value <math>-\infty</math>, the continuous functions can be taken to be real-valued.<ref>Stromberg, p. 132, Exercise 4(g)</ref><ref>{{cite web |title=Show that lower semicontinuous function is the supremum of an increasing sequence of continuous functions |url=https://math.stackexchange.com/q/1279763}}</ref>

:Additionally, every upper semicontinuous function <math>f:X\to\overline{\R}</math> is the limit of a [[monotone decreasing]] sequence of extended real-valued continuous functions on <math>X;</math> if <math>f</math> does not take the value <math>\infty,</math> the continuous functions can be taken to be real-valued.

* Any upper semicontinuous function <math>f : X \to \N</math> on an arbitrary topological space <math>X</math> is locally constant on some [[dense set|dense open subset]] of <math>X.</math>

* If the topological space <math>X</math> is [[Sequential_space|sequential]], then <math>f : X \to \mathbb{R}</math> is upper semi-continuous if and only if it is sequentially upper semi-continuous, that is, if for any <math>x \in X</math> and any sequence <math>(x_n)_n \subset X</math> that converges towards <math>x</math>, there holds <math>\limsup_{n \to \infty} f(x_n) \leqslant f(x)</math>. Equivalently, in a sequential space, <math>f</math> is upper semicontinuous if and only if its superlevel sets <math>\{\, x \in X \,|\, f(x) \geqslant y \,\}</math> are [[Fréchet–Urysohn_space#Sequentially_open%2Fclosed_sets|sequentially closed]] for all <math>y \in \mathbb{R}</math>. In general, upper semicontinuous functions are sequentially upper semicontinuous, but the converse may be false.