Definitions
Assume throughout that
is a topological space and
is a function with values in the extended real numbers
.
Upper semicontinuity
A function
is called upper semicontinuous at a point
if for every real
there exists a neighborhood
of
such that
for all
.[2]
Equivalently,
is upper semicontinuous at
if and only if
![{\displaystyle \limsup _{x\to x_{0))f(x)\leq f(x_{0})}](https://wikimedia.org/api/rest_v1/media/math/render/svg/e6bb388e034b1c471b1e26add49448bc64159714)
where lim sup is the limit superior of the function
at the point
.
A function
is called upper semicontinuous if it satisfies any of the following equivalent conditions:[2]
- (1) The function is upper semicontinuous at every point of its domain.
- (2) All sets
with
are open in
, where
.
- (3) All superlevel sets
with
are closed in
.
- (4) The hypograph
is closed in
.
- (5) The function is continuous when the codomain
is given the left order topology. This is just a restatement of condition (2) since the left order topology is generated by all the intervals
.
Lower semicontinuity
A function
is called lower semicontinuous at a point
if for every real
there exists a neighborhood
of
such that
for all
.
Equivalently,
is lower semicontinuous at
if and only if
![{\displaystyle \liminf _{x\to x_{0))f(x)\geq f(x_{0})}](https://wikimedia.org/api/rest_v1/media/math/render/svg/26435cc30871de2f122852ea40398177194aa0b2)
where
is the limit inferior of the function
at point
.
A function
is called lower semicontinuous if it satisfies any of the following equivalent conditions:
- (1) The function is lower semicontinuous at every point of its domain.
- (2) All sets
with
are open in
, where
.
- (3) All sublevel sets
with
are closed in
.
- (4) The epigraph
is closed in
.
- (5) The function is continuous when the codomain
is given the right order topology. This is just a restatement of condition (2) since the right order topology is generated by all the intervals
.
Examples
Consider the function
piecewise defined by:
![{\displaystyle f(x)={\begin{cases}-1&{\mbox{if ))x<0,\\1&{\mbox{if ))x\geq 0\end{cases))}](https://wikimedia.org/api/rest_v1/media/math/render/svg/9bf20d6adfc01c674e2d18877732cc06aa8741c0)
This function is upper semicontinuous at
but not lower semicontinuous.
The floor function
which returns the greatest integer less than or equal to a given real number
is everywhere upper semicontinuous. Similarly, the ceiling function
is lower semicontinuous.
Upper and lower semicontinuity bear no relation to 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.[3] For example the function
![{\displaystyle f(x)={\begin{cases}\sin(1/x)&{\mbox{if ))x\neq 0,\\1&{\mbox{if ))x=0,\end{cases))}](https://wikimedia.org/api/rest_v1/media/math/render/svg/8baee1f6ab697f33589bf6b28c8c5a3fd11fd68e)
is upper semicontinuous at
while the function limits from the left or right at zero do not even exist.
If
is a Euclidean space (or more generally, a metric space) and
is the space of curves in
(with the supremum distance
), then the length functional
which assigns to each curve
its length
is lower semicontinuous.[4] 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
.
Let
be a measure space and let
denote the set of positive measurable functions endowed with the
topology of convergence in measure with respect to
Then by Fatou's lemma the integral, seen as an operator from
to
is lower semicontinuous.
Properties
Unless specified otherwise, all functions below are from a topological space
to the extended real numbers
Several of the results hold for semicontinuity at a specific point, but for brevity they are only stated for semicontinuity over the whole domain.
- A function
is continuous if and only if it is both upper and lower semicontinuous.
- The indicator function of a set
(defined by
if
and
if
) is upper semicontinuous if and only if
is a closed set. It is lower semicontinuous if and only if
is an open set.[note 1]
- The sum
of two lower semicontinuous functions is lower semicontinuous[5] (provided the sum is well-defined, i.e.,
is not the indeterminate form
). The same holds for upper semicontinuous functions.
- If both functions are non-negative, the product function
of two lower semicontinuous functions is lower semicontinuous. The corresponding result holds for upper semicontinuous functions.
- A function
is lower semicontinuous if and only if
is upper semicontinuous.
- The composition
of upper semicontinuous functions is not necessarily upper semicontinuous, but if
is also non-decreasing, then
is upper semicontinuous.[6]
- The minimum and the maximum of two lower semicontinuous functions are lower semicontinuous. In other words, the set of all lower semicontinuous functions from
to
(or to
) forms a lattice. The same holds for upper semicontinuous functions.
- The (pointwise) supremum of an arbitrary family
of lower semicontinuous functions
(defined by
) is lower semicontinuous.[7]
- In particular, the limit of a monotone increasing sequence
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
defined for
for ![{\displaystyle n=1,2,\ldots .}](https://wikimedia.org/api/rest_v1/media/math/render/svg/7232c7585f99f91de353d07f3562bb372b94393b)
- 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.
- (Theorem of Baire)[note 2] Assume
is a metric space. Every lower semicontinuous function
is the limit of a monotone increasing sequence of extended real-valued continuous functions on
; if
does not take the value
, the continuous functions can be taken to be real-valued.[8][9]
- And every upper semicontinuous function
is the limit of a monotone decreasing sequence of extended real-valued continuous functions on
; if
does not take the value
the continuous functions can be taken to be real-valued.
- If
is a compact space (for instance a closed bounded interval
) and
is upper semicontinuous, then
has a maximum on
If
is lower semicontinuous on
it has a minimum on ![{\displaystyle C.}](https://wikimedia.org/api/rest_v1/media/math/render/svg/067be67e68f60c53ce83241748d0d6249675c58d)
- (Proof for the upper semicontinuous case: By condition (5) in the definition,
is continuous when
is given the left order topology. So its image
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.)