# Semi-continuity

In mathematical analysis, **semi-continuity** (or **semicontinuity**) is a property of extended real-valued functions that is weaker than continuity. An extended real-valued function *f* is **upper** (respectively, **lower**) **semi-continuous** at a point *x*_{0} if, roughly speaking, the function values for arguments near *x*_{0} are either close to *f*(*x*_{0}) or less than (respectively, greater than) *f*(*x*_{0}).

## Examples[edit]

Consider the function *f*, piecewise defined by *f*(*x*) = –1 for *x* < 0 and *f*(*x*) = 1 for *x* ≥ 0; this function is upper semi-continuous at *x*_{0} = 0, but not lower semi-continuous.

The indicator function of a closed set is upper semi-continuous, whereas the indicator function of an open set is lower semi-continuous; the floor function , which returns the greatest integer less than or equal to a given real number *x*, is everywhere upper semi-continuous. Similarly, the ceiling function is lower semi-continuous.

A function may be upper or lower semi-continuous without being either left or right continuous. For example, the function

is upper semi-continuous at *x* = 1 although not left or right continuous. The limit from the left is equal to 1 and the limit from the right is equal to 1/2, both of which are different from the function value of 2. Similarly the function

is upper semi-continuous at *x* = 0 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.

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 semi-continuous.

## Formal definition[edit]

Suppose is a topological space, is a point in and is an extended real-valued function.

We say that is **upper semi-continuous** at if for every there exists a neighborhood of such that for all when , and tends to as tends towards when .

For the particular case of a metric space, this can be expressed as

where lim sup is the limit superior (of the function at point ). (For non-metric spaces, an equivalent definition using nets may be stated.)

The function is called upper semi-continuous if it is upper semi-continuous at every point of its domain. A function is upper semi-continuous if and only if is an open set for every .

We say that is **lower semi-continuous** at if for every there exists a neighborhood of such that for all in when , and tends to as tends towards when . Equivalently, in the case of a metric space, this can be expressed as

where is the limit inferior (of the function at point ).

The function *f* is called lower semi-continuous if it is lower semi-continuous at every point of its domain. A function is lower semi-continuous if and only if is an open set for every α ∈ **R**; alternatively, a function is lower semi-continuous if and only if all of its lower level sets are closed. Lower level sets are also called *sublevel sets* or *trenches*.^{[1]}

## Properties[edit]

A function is continuous at *x*_{0} if and only if it is upper and lower semi-continuous there. Therefore, semi-continuity can be used to prove continuity.

If *f* and *g* are two real-valued functions which are both upper semi-continuous at *x*_{0}, then so is *f* + *g*. If both functions are non-negative, then the product function *fg* will also be upper semi-continuous at *x*_{0}. The same holds for functions lower semi-continuous at *x*_{0}.^{[2]}

The composition *f*∘*g* of upper semi-continuous functions *f* and *g* is not necessarily upper semi-continuous, but if *f* is also non-decreasing, then *f*∘*g* is upper semi-continuous.^{[3]}

Multiplying a positive upper semi-continuous function with a negative number turns it into a lower semi-continuous function.

If *C* is a compact space (for instance a closed, bounded interval [*a*, *b*]) and *f* : *C* → [–∞,∞) is upper semi-continuous, then *f* has a maximum on *C*; the analogous statement for (–∞,∞]-valued lower semi-continuous functions and minima is also true. (See the article on the extreme value theorem for a proof.)

Suppose *f*_{i} : *X* → [–∞,∞] is a lower semi-continuous function for every index *i* in a nonempty set *I*, and define *f* as pointwise supremum, i.e.,

Then *f* is lower semi-continuous^{[4]}. Even if all the *f*_{i} are continuous, *f* need not be continuous: indeed every lower semi-continuous function on a uniform space (e.g. a metric space) arises as the supremum of a sequence of continuous functions.

Likewise, the pointwise infimum of an arbitrary collection of upper semicontinuous functions is upper semicontinuous.

The indicator function of any open set is lower semicontinuous; the indicator function of a closed set is upper semicontinuous. However, in convex analysis, the term "indicator function" often refers to the characteristic function, and the characteristic function of any *closed* set is lower semicontinuous, and the characteristic function of any *open* set is upper semicontinuous.

A function *f* : **R**^{n}→**R** is lower semicontinuous if and only if its epigraph (the set of points lying on or above its graph) is closed.

A function *f* : *X*→**R**, for some topological space *X*, is lower semicontinuous if and only if it is continuous with respect to the Scott topology on **R**.

Any upper semicontinuous function *f* : *X*→**N** on an arbitrary topological space *X* is locally constant on some dense open subset of *X*.

The maximum and minimum of finitely many upper semicontinuous functions is upper semicontinuous, and the same holds true of lower semicontinuous functions.

## See also[edit]

## References[edit]

**^**Kiwiel, Krzysztof C. (2001). "Convergence and efficiency of subgradient methods for quasiconvex minimization".*Mathematical Programming, Series A*.**90**(1). Berlin, Heidelberg: Springer. pp. 1–25. doi:10.1007/PL00011414. ISSN 0025-5610. MR 1819784.**^**Puterman, Martin L. (2005).*Markov Decision Processes Discrete Stochastic Dynamic Programming*. Wiley-Interscience. p. 602. ISBN 978-0-471-72782-8.**^**Moore, James C. (1999).*Mathematical methods for economic theory*. Berlin: Springer. p. 143. ISBN 9783540662358.**^**"Baire theorem".*Encyclopedia of Mathematics*.

## Further reading[edit]

- Benesova, B.; Kruzik, M. (2017). "Weak Lower Semicontinuity of Integral Functionals and Applications".
*SIAM Review*.**59**(4): 703–766. arXiv:1601.00390. doi:10.1137/16M1060947. - Bourbaki, Nicolas (1998).
*Elements of Mathematics: General Topology, 1–4*. Springer. ISBN 0-201-00636-7. - Bourbaki, Nicolas (1998).
*Elements of Mathematics: General Topology, 5–10*. Springer. ISBN 3-540-64563-2. - Gelbaum, Bernard R.; Olmsted, John M.H. (2003).
*Counterexamples in analysis*. Dover Publications. ISBN 0-486-42875-3. - Hyers, Donald H.; Isac, George; Rassias, Themistocles M. (1997).
*Topics in nonlinear analysis & applications*. World Scientific. ISBN 981-02-2534-2.