Editing Semi-continuity (section)

== Examples ==
Consider the function <math>f,</math> [[piecewise]] defined by:
<math display=block>f(x) = \begin{cases}
-1   & \mbox{if } x < 0,\\
 1   & \mbox{if } x \geq 0
\end{cases}</math>
This function is upper semicontinuous at <math>x_0 = 0,</math> but not lower semicontinuous.

The [[floor function]] <math>f(x) = \lfloor x \rfloor,</math> which returns the greatest integer less than or equal to a given real number <math>x,</math> is everywhere upper semicontinuous. Similarly, the [[ceiling function]] <math>f(x) = \lceil x \rceil</math> is lower semicontinuous.

Upper and lower semicontinuity bear no relation to [[Continuous function#Directional and semi-continuity|continuity from the left or from the right]] for functions of a real variable. Semicontinuity is defined in terms of an ordering in the range of the functions, not in the domain.<ref>Willard, p. 49, problem 7K</ref>  For example the function
<math display=block>f(x) = \begin{cases}
\sin(1/x) & \mbox{if } x \neq 0,\\
1         & \mbox{if } x = 0,
\end{cases}</math>
is upper semicontinuous at <math>x = 0</math> while the function limits from the left or right at zero do not even exist.

If <math>X = \R^n</math> is a Euclidean space (or more generally, a metric space) and <math>\Gamma = C([0,1], X)</math> is the space of [[curve]]s in <math>X</math> (with the [[Supremum norm|supremum distance]] <math>d_\Gamma(\alpha,\beta) = \sup\{d_X(\alpha(t),\beta(t)):t\in[0,1]\}</math>), then the length functional <math>L : \Gamma \to [0, +\infty],</math> which assigns to each curve <math>\alpha</math> its [[Curve#Length of curves|length]] <math>L(\alpha),</math> is lower semicontinuous.<ref>{{Cite book |last=Giaquinta |first=Mariano |url=https://www.worldcat.org/oclc/213079540 |title=Mathematical analysis : linear and metric structures and continuity |date=2007 |publisher=Birkhäuser |others=Giuseppe Modica |isbn=978-0-8176-4514-4 |edition=1 |location=Boston |at=Theorem 11.3, p.396 |oclc=213079540}}</ref> As an example, consider approximating the unit square diagonal by a staircase from below. The staircase always has length 2, while the diagonal line has only length <math>\sqrt 2</math>.