Semi-continuity

For the notion of upper or lower semi-continuous set-valued function, see Hemicontinuity.

In mathematical analysis, semicontinuity (or semi-continuity) is a property of extended real-valued functions that is weaker than continuity. An extended real-valued function ${\displaystyle f}$ is upper (respectively, lower) semicontinuous at a point ${\displaystyle x_{0}}$ if, roughly speaking, the function values for arguments near ${\displaystyle x_{0}}$ are not much higher (respectively, lower) than ${\displaystyle f\left(x_{0}\right).}$

A function is continuous if and only if it is both upper and lower semicontinuous. If we take a continuous function and increase its value at a certain point ${\displaystyle x_{0}}$ to ${\displaystyle f\left(x_{0}\right)+c}$ for some ${\displaystyle c>0}$, then the result is upper semicontinuous; if we decrease its value to ${\displaystyle f\left(x_{0}\right)-c}$ then the result is lower semicontinuous.

A function that is upper semicontinuous, but not lower semicontinuous, at ${\displaystyle x_{0}}$. The solid blue dot indicates ${\displaystyle f\left(x_{0}\right).}$

A function that is lower semicontinuous, but not upper semicontinuous, at ${\displaystyle x_{0}}$. The solid blue dot indicates ${\displaystyle f\left(x_{0}\right).}$

The notion of upper and lower semicontinuous function was first introduced and studied by René Baire in his thesis in 1899. ^[1]

Definitions

Assume throughout that ${\displaystyle X}$ is a topological space and ${\displaystyle f:X\to {\overline {\mathbb {R} }}}$ is a function with values in the extended real numbers ${\displaystyle {\overline {\mathbb {R} }}=\mathbb {R} \cup \{-\infty ,\infty \}=[-\infty ,\infty ]}$.

Upper semicontinuity

A function ${\displaystyle f:X\to {\overline {\mathbb {R} }}}$ is called upper semicontinuous at a point${\displaystyle x_{0}\in X}$ if for every real ${\displaystyle y>f\left(x_{0}\right)}$ there exists a neighborhood ${\displaystyle U}$ of ${\displaystyle x_{0}}$ such that ${\displaystyle f(x)<y}$ for all ${\displaystyle x\in U}$. ^[2] Equivalently, ${\displaystyle f}$ is upper semicontinuous at ${\displaystyle x_{0}}$ if and only if $${\displaystyle \limsup _{x\to x_{0}}f(x)\leq f(x_{0})}$$ where lim sup is the limit superior of the function ${\displaystyle f}$ at the point ${\displaystyle x_{0}.}$

If ${\displaystyle X}$ is a metric space with distance function ${\displaystyle d}$ and ${\displaystyle f(x_{0})\in \mathbb {R} ,}$ this can also be restated using an ${\displaystyle \varepsilon }$-${\displaystyle \delta }$ formulation, similar to the definition of continuous function. Namely, for each ${\displaystyle \varepsilon >0}$ there is a ${\displaystyle \delta >0}$ such that ${\displaystyle f(x)<f(x_{0})+\varepsilon }$ whenever ${\displaystyle d(x,x_{0})<\delta .}$

A function ${\displaystyle f:X\to {\overline {\mathbb {R} }}}$ 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) For each ${\displaystyle y\in \mathbb {R} }$, the set ${\displaystyle f^{-1}([-\infty ,y))=\{x\in X:f(x)<y\}}$ is open in ${\displaystyle X}$, where ${\displaystyle [-\infty ,y)=\{t\in {\overline {\mathbb {R} }}:t<y\}}$.

(3) For each ${\displaystyle y\in \mathbb {R} }$, the ${\displaystyle y}$-superlevel set ${\displaystyle f^{-1}([y,\infty ))=\{x\in X:f(x)\geq y\}}$ is closed in ${\displaystyle X}$.

(4) The hypograph ${\displaystyle \{(x,t)\in X\times \mathbb {R} :t\leq f(x)\}}$ is closed in ${\displaystyle X\times \mathbb {R} }$.

(5) The function ${\displaystyle f}$ is continuous when the codomain ${\displaystyle {\overline {\mathbb {R} }}}$ 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 ${\displaystyle [-\infty ,y)}$.

Lower semicontinuity

A function ${\displaystyle f:X\to {\overline {\mathbb {R} }}}$ is called lower semicontinuous at a point${\displaystyle x_{0}\in X}$ if for every real ${\displaystyle y<f\left(x_{0}\right)}$ there exists a neighborhood ${\displaystyle U}$ of ${\displaystyle x_{0}}$ such that ${\displaystyle f(x)>y}$ for all ${\displaystyle x\in U}$. Equivalently, ${\displaystyle f}$ is lower semicontinuous at ${\displaystyle x_{0}}$ if and only if $${\displaystyle \liminf _{x\to x_{0}}f(x)\geq f(x_{0})}$$ where ${\displaystyle \liminf }$ is the limit inferior of the function ${\displaystyle f}$ at point ${\displaystyle x_{0}.}$

If ${\displaystyle X}$ is a metric space with distance function ${\displaystyle d}$ and ${\displaystyle f(x_{0})\in \mathbb {R} ,}$ this can also be restated as follows: For each ${\displaystyle \varepsilon >0}$ there is a ${\displaystyle \delta >0}$ such that ${\displaystyle f(x)>f(x_{0})-\varepsilon }$ whenever ${\displaystyle d(x,x_{0})<\delta .}$

A function ${\displaystyle f:X\to {\overline {\mathbb {R} }}}$ 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) For each ${\displaystyle y\in \mathbb {R} }$, the set ${\displaystyle f^{-1}((y,\infty ])=\{x\in X:f(x)>y\}}$ is open in ${\displaystyle X}$, where ${\displaystyle (y,\infty ]=\{t\in {\overline {\mathbb {R} }}:t>y\}}$.

(3) For each ${\displaystyle y\in \mathbb {R} }$, the ${\displaystyle y}$-sublevel set ${\displaystyle f^{-1}((-\infty ,y])=\{x\in X:f(x)\leq y\}}$ is closed in ${\displaystyle X}$.

(4) The epigraph ${\displaystyle \{(x,t)\in X\times \mathbb {R} :t\geq f(x)\}}$ is closed in ${\displaystyle X\times \mathbb {R} }$. ^{[3] : 207}

(5) The function ${\displaystyle f}$ is continuous when the codomain ${\displaystyle {\overline {\mathbb {R} }}}$ 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 ${\displaystyle (y,\infty ]}$.

Examples

Consider the function ${\displaystyle f,}$ piecewise defined by: $${\displaystyle f(x)={\begin{cases}-1&{\mbox{if }}x<0,\\1&{\mbox{if }}x\geq 0\end{cases}}}$$ This function is upper semicontinuous at ${\displaystyle x_{0}=0,}$ but not lower semicontinuous.

The floor function ${\displaystyle f(x)=\lfloor x\rfloor ,}$ which returns the greatest integer less than or equal to a given real number ${\displaystyle x,}$ is everywhere upper semicontinuous. Similarly, the ceiling function ${\displaystyle f(x)=\lceil x\rceil }$ 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. ^[4] For example the function $${\displaystyle f(x)={\begin{cases}\sin(1/x)&{\mbox{if }}x\neq 0,\\1&{\mbox{if }}x=0,\end{cases}}}$$ is upper semicontinuous at ${\displaystyle x=0}$ while the function limits from the left or right at zero do not even exist.

If ${\displaystyle X=\mathbb {R} ^{n}}$ is a Euclidean space (or more generally, a metric space) and ${\displaystyle \Gamma =C([0,1],X)}$ is the space of curves in ${\displaystyle X}$ (with the supremum distance ${\displaystyle d_{\Gamma }(\alpha ,\beta )=\sup\{d_{X}(\alpha (t),\beta (t)):t\in [0,1]\}}$), then the length functional ${\displaystyle L:\Gamma \to [0,+\infty ],}$ which assigns to each curve ${\displaystyle \alpha }$ its length ${\displaystyle L(\alpha ),}$ is lower semicontinuous. ^[5] 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 ${\displaystyle {\sqrt {2}}}$.

Let ${\displaystyle (X,\mu )}$ be a measure space and let ${\displaystyle L^{+}(X,\mu )}$ denote the set of positive measurable functions endowed with the topology of convergence in measure with respect to ${\displaystyle \mu .}$ Then by Fatou's lemma the integral, seen as an operator from ${\displaystyle L^{+}(X,\mu )}$ to ${\displaystyle [-\infty ,+\infty ]}$ is lower semicontinuous.

Tonelli's theorem in functional analysis characterizes the weak lower semicontinuity of nonlinear functionals on L^p spaces in terms of the convexity of another function.

Properties

Unless specified otherwise, all functions below are from a topological space ${\displaystyle X}$ to the extended real numbers ${\displaystyle {\overline {\mathbb {R} }}=[-\infty ,\infty ].}$ 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 ${\displaystyle f:X\to {\overline {\mathbb {R} }}}$ is continuous if and only if it is both upper and lower semicontinuous.
• The characteristic function or indicator function of a set ${\displaystyle A\subset X}$ (defined by ${\displaystyle \mathbf {1} _{A}(x)=1}$ if ${\displaystyle x\in A}$ and ${\displaystyle 0}$ if ${\displaystyle x\notin A}$) is upper semicontinuous if and only if ${\displaystyle A}$ is a closed set. It is lower semicontinuous if and only if ${\displaystyle A}$ is an open set.
• In the field of convex analysis, the characteristic function of a set ${\displaystyle A\subset X}$ is defined differently, as ${\displaystyle \chi _{A}(x)=0}$ if ${\displaystyle x\in A}$ and ${\displaystyle \chi _{A}(x)=\infty }$ if ${\displaystyle x\notin A}$. With that definition, the characteristic function of any closed set is lower semicontinuous, and the characteristic function of any open set is upper semicontinuous.

Binary Operations on Semicontinuous Functions

Let ${\displaystyle f,g:X\to {\overline {\mathbb {R} }}}$.

• If ${\displaystyle f}$ and ${\displaystyle g}$ are lower semicontinuous, then the sum ${\displaystyle f+g}$ is lower semicontinuous ^[6] (provided the sum is well-defined, i.e., ${\displaystyle f(x)+g(x)}$ is not the indeterminate form ${\displaystyle -\infty +\infty }$). The same holds for upper semicontinuous functions.
• If ${\displaystyle f}$ and ${\displaystyle g}$ are lower semicontinuous and non-negative, then the product function ${\displaystyle fg}$ is lower semicontinuous. The corresponding result holds for upper semicontinuous functions.
• The function ${\displaystyle f}$ is lower semicontinuous if and only if ${\displaystyle -f}$ is upper semicontinuous.
• If ${\displaystyle f}$ and ${\displaystyle g}$ are upper semicontinuous and ${\displaystyle f}$ is non-decreasing, then the composition ${\displaystyle f\circ g}$ is upper semicontinuous. On the other hand, if ${\displaystyle f}$ is not non-decreasing, then ${\displaystyle f\circ g}$ may not be upper semicontinuous. ^[7]
• If ${\displaystyle f}$ and ${\displaystyle g}$ are lower semicontinuous, their (pointwise) maximum and minimum (defined by ${\displaystyle x\mapsto \max\{f(x),g(x)\}}$ and ${\displaystyle x\mapsto \min\{f(x),g(x)\}}$) are also lower semicontinuous. Consequently, the set of all lower semicontinuous functions from ${\displaystyle X}$ to ${\displaystyle {\overline {\mathbb {R} }}}$ (or to ${\displaystyle \mathbb {R} }$) forms a lattice. The corresponding statements also hold for upper semicontinuous functions.

Optimization of Semicontinuous Functions

• The (pointwise) supremum of an arbitrary family ${\displaystyle (f_{i})_{i\in I}}$ of lower semicontinuous functions ${\displaystyle f_{i}:X\to {\overline {\mathbb {R} }}}$ (defined by ${\displaystyle f(x)=\sup\{f_{i}(x):i\in I\}}$) is lower semicontinuous. ^[8]

In particular, the limit of a monotone increasing sequence ${\displaystyle f_{1}\leq f_{2}\leq f_{3}\leq \cdots }$ 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 ${\displaystyle f_{n}(x)=1-(1-x)^{n}}$ defined for ${\displaystyle x\in [0,1]}$ for ${\displaystyle n=1,2,\ldots .}$

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.

• If ${\displaystyle C}$ is a compact space (for instance a closed bounded interval ${\displaystyle [a,b]}$) and ${\displaystyle f:C\to {\overline {\mathbb {R} }}}$ is upper semicontinuous, then ${\displaystyle f}$ attains a maximum on ${\displaystyle C.}$ If ${\displaystyle f}$ is lower semicontinuous on ${\displaystyle C,}$ it attains a minimum on ${\displaystyle C.}$

(Proof for the upper semicontinuous case: By condition (5) in the definition, ${\displaystyle f}$ is continuous when ${\displaystyle {\overline {\mathbb {R} }}}$ is given the left order topology. So its image ${\displaystyle f(C)}$ 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.)

Other Properties

• (Theorem of Baire) ^{[note 1]} Let ${\displaystyle X}$ be a metric space. Every lower semicontinuous function ${\displaystyle f:X\to {\overline {\mathbb {R} }}}$ is the limit of a point-wise increasing sequence of extended real-valued continuous functions on ${\displaystyle X.}$ In particular, there exists a sequence ${\displaystyle \{f_{i}\}}$ of continuous functions ${\displaystyle f_{i}:X\to {\overline {\mathbb {R} }}}$ such that

$${\displaystyle f_{i}(x)\leq f_{i+1}(x)\quad \forall x\in X,\ \forall i=0,1,2,\dots }$$ and

$${\displaystyle \lim _{i\to \infty }f_{i}(x)=f(x)\quad \forall x\in X.}$$

If ${\displaystyle f}$ does not take the value ${\displaystyle -\infty }$, the continuous functions can be taken to be real-valued. ^{[9] [10]}

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

• Any upper semicontinuous function ${\displaystyle f:X\to \mathbb {N} }$ on an arbitrary topological space ${\displaystyle X}$ is locally constant on some dense open subset of ${\displaystyle X.}$

• If the topological space ${\displaystyle X}$ is sequential, then ${\displaystyle f:X\to \mathbb {R} }$ is upper semi-continuous if and only if it is sequentially upper semi-continuous, that is, if for any ${\displaystyle x\in X}$ and any sequence ${\displaystyle (x_{n})_{n}\subset X}$ that converges towards ${\displaystyle x}$, there holds ${\displaystyle \limsup _{n\to \infty }f(x_{n})\leqslant f(x)}$. Equivalently, in a sequential space, ${\displaystyle f}$ is upper semicontinuous if and only if its superlevel sets ${\displaystyle \{\,x\in X\,|\,f(x)\geqslant y\,\}}$ are sequentially closed for all ${\displaystyle y\in \mathbb {R} }$. In general, upper semicontinuous functions are sequentially upper semicontinuous, but the converse may be false.

Semicontinuity of Set-valued Functions

For set-valued functions, several concepts of semicontinuity have been defined, namely upper, lower, outer, and inner semicontinuity, as well as upper and lower hemicontinuity . A set-valued function ${\displaystyle F}$ from a set ${\displaystyle A}$ to a set ${\displaystyle B}$ is written ${\displaystyle F:A\rightrightarrows B.}$ For each ${\displaystyle x\in A,}$ the function ${\displaystyle F}$ defines a set ${\displaystyle F(x)\subset B.}$ The preimage of a set ${\displaystyle S\subset B}$ under ${\displaystyle F}$ is defined as $${\displaystyle F^{-1}(S):=\{x\in A:F(x)\cap S\neq \varnothing \}.}$$ That is, ${\displaystyle F^{-1}(S)}$ is the set that contains every point ${\displaystyle x}$ in ${\displaystyle A}$ such that ${\displaystyle F(x)}$ is not disjoint from ${\displaystyle S}$. ^[11]

Upper and Lower Semicontinuity

A set-valued map ${\displaystyle F:\mathbb {R} ^{m}\rightrightarrows \mathbb {R} ^{n}}$ is upper semicontinuous at ${\displaystyle x\in \mathbb {R} ^{m}}$ if for every open set ${\displaystyle U\subset \mathbb {R} ^{n}}$ such that ${\displaystyle F(x)\subset U}$, there exists a neighborhood ${\displaystyle V}$ of ${\displaystyle x}$ such that ${\displaystyle F(V)\subset U.}$ ^{[11] : Def. 2.1}

A set-valued map ${\displaystyle F:\mathbb {R} ^{m}\rightrightarrows \mathbb {R} ^{n}}$ is lower semicontinuous at ${\displaystyle x\in \mathbb {R} ^{m}}$ if for every open set ${\displaystyle U\subset \mathbb {R} ^{n}}$ such that ${\displaystyle x\in F^{-1}(U),}$ there exists a neighborhood ${\displaystyle V}$ of ${\displaystyle x}$ such that ${\displaystyle V\subset F^{-1}(U).}$ ^{[11] : Def. 2.2}

Upper and lower set-valued semicontinuity are also defined more generally for a set-valued maps between topological spaces by replacing ${\displaystyle \mathbb {R} ^{m}}$ and ${\displaystyle \mathbb {R} ^{n}}$ in the above definitions with arbitrary topological spaces. ^[11]

Note, that there is not a direct correspondence between single-valued lower and upper semicontinuity and set-valued lower and upper semicontinuouty. An upper semicontinuous single-valued function is not necessarily upper semicontinuous when considered as a set-valued map. ^{[11] : 18} For example, the function ${\displaystyle f:\mathbb {R} \to \mathbb {R} }$ defined by $${\displaystyle f(x)={\begin{cases}-1&{\mbox{if }}x<0,\\1&{\mbox{if }}x\geq 0\end{cases}}}$$ is upper semicontinuous in the single-valued sense but the set-valued map ${\displaystyle x\mapsto F(x):=\{f(x)\}}$ is not upper semicontinuous in the set-valued sense.

Inner and Outer Semicontinuity

A set-valued function ${\displaystyle F:\mathbb {R} ^{m}\rightrightarrows \mathbb {R} ^{n}}$ is called inner semicontinuous at ${\displaystyle x}$ if for every ${\displaystyle y\in F(x)}$ and every convergent sequence ${\displaystyle (x_{i})}$ in ${\displaystyle \mathbb {R} ^{m}}$ such that ${\displaystyle x_{i}\to x}$, there exists a sequence ${\displaystyle (y_{i})}$ in ${\displaystyle \mathbb {R} ^{n}}$ such that ${\displaystyle y_{i}\to y}$ and ${\displaystyle y_{i}\in F\left(x_{i}\right)}$ for all sufficiently large ${\displaystyle i\in \mathbb {N} .}$ ^{[12] [note 2]}

A set-valued function ${\displaystyle F:\mathbb {R} ^{m}\rightrightarrows \mathbb {R} ^{n}}$ is called outer semicontinuous at ${\displaystyle x}$ if for every convergence sequence ${\displaystyle (x_{i})}$ in ${\displaystyle \mathbb {R} ^{m}}$ such that ${\displaystyle x_{i}\to x}$ and every convergent sequence ${\displaystyle (y_{i})}$ in ${\displaystyle \mathbb {R} ^{n}}$ such that ${\displaystyle y_{i}\in F(x_{i})}$ for each ${\displaystyle i\in \mathbb {N} ,}$ the sequence ${\displaystyle (y_{i})}$ converges to a point in ${\displaystyle F(x)}$ (that is, ${\displaystyle \lim _{i\to \infty }y_{i}\in F(x)}$). ^[12]

See also

• Directional continuity  – Mathematical function with no sudden changesPages displaying short descriptions of redirect targets
• Katětov–Tong insertion theorem  – On existence of a continuous function between semicontinuous upper and lower bounds
• Hemicontinuity  – Semicontinuity for set-valued functions
• Càdlàg  – Right continuous function with left limits

Notes

1. The result was proved by René Baire in 1904 for real-valued function defined on ${\displaystyle \mathbb {R} }$. It was extended to metric spaces by 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 spaces. (See Engelking, Exercise 1.7.15(c), p. 62 for details and specific references.)
2. In particular, there exists ${\displaystyle i_{0}\geq 0}$ such that ${\displaystyle y_{i}\in F(x_{i})}$ for every natural number ${\displaystyle i\geq i_{0},}$. The necessisty of only considering the tail of ${\displaystyle y_{i}}$ comes from the fact that for small values of ${\displaystyle i,}$ the set ${\displaystyle F(x_{i})}$ may be empty.

References

1. Verry, Matthieu. "Histoire des mathématiques - René Baire".
2. Stromberg, p. 132, Exercise 4
3. Kurdila, A. J., Zabarankin, M. (2005). "Convex Functional Analysis". Lower Semicontinuous Functionals. Systems & Control: Foundations & Applications (1st ed.). Birkhäuser-Verlag. pp. 205–219. doi:10.1007/3-7643-7357-1_7. ISBN   978-3-7643-2198-7.
4. Willard, p. 49, problem 7K
5. Giaquinta, Mariano (2007). Mathematical analysis : linear and metric structures and continuity. Giuseppe Modica (1 ed.). Boston: Birkhäuser. Theorem 11.3, p.396. ISBN   978-0-8176-4514-4. OCLC   213079540.
6. Puterman, Martin L. (2005). Markov Decision Processes Discrete Stochastic Dynamic Programming . Wiley-Interscience. pp.  602. ISBN   978-0-471-72782-8.
7. Moore, James C. (1999). Mathematical methods for economic theory . Berlin: Springer. p.  143. ISBN   9783540662358.
8. "To show that the supremum of any collection of lower semicontinuous functions is lower semicontinuous".
9. Stromberg, p. 132, Exercise 4(g)
10. "Show that lower semicontinuous function is the supremum of an increasing sequence of continuous functions".
11. Freeman, R. A., Kokotović, P. (1996). Robust Nonlinear Control Design. Birkhäuser Boston. doi:10.1007/978-0-8176-4759-9. ISBN   978-0-8176-4758-2..
12. Goebel, R. K. (January 2024). "Set-Valued, Convex, and Nonsmooth Analysis in Dynamics and Control: An Introduction". Chapter 2: Set convergence and set-valued mappings. Other Titles in Applied Mathematics. Society for Industrial and Applied Mathematics. pp. 21–36. doi:10.1137/1.9781611977981.ch2. ISBN   978-1-61197-797-4.

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