Decreasing Sequence of Continuous Functions Converging to an Upper Semi Continuous Function

Property of functions which is weaker than continuity

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.

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

A lower semicontinuous function that is not upper semicontinuous. 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 [edit]

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 [edit]

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 ${\displaystyle x_{0}}$ .

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) All sets ${\displaystyle f^{-1}((\leftarrow ,y))=\{x\in X:f(x)<y\}}$ with ${\displaystyle y\in \mathbb {R} }$ are open in ${\displaystyle X}$ , where ${\displaystyle (\leftarrow ,y)=\{t\in {\overline {\mathbb {R} }}:t<y\}}$ .

(3) All superlevel sets ${\displaystyle \{x\in X:f(x)\geq y\}}$ with ${\displaystyle y\in \mathbb {R} }$ are 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 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 ${\displaystyle (\leftarrow ,y)}$ .

Lower semicontinuity [edit]

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 ${\displaystyle x_{0}}$ .

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) All sets ${\displaystyle f^{-1}((y,\rightarrow ))=\{x\in X:f(x)>y\}}$ with ${\displaystyle y\in \mathbb {R} }$ are open in ${\displaystyle X}$ , where ${\displaystyle (y,\rightarrow )=\{t\in {\overline {\mathbb {R} }}:t>y\}}$ .

(3) All sublevel sets ${\displaystyle \{x\in X:f(x)\leq y\}}$ with ${\displaystyle y\in \mathbb {R} }$ are closed in ${\displaystyle X}$ .

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

(5) The function 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 ${\displaystyle (y,\rightarrow )}$ .

Examples [edit]

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.^[3] For example the function

$${\displaystyle f(x)={\begin{cases}\sin(1/x)&{\mbox{if }}x\neq 0,\\1&{\mbox{if }}x=0,\end{cases}}}$$

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.^[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 ${\displaystyle {\sqrt {2}}}$ .

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.

Properties [edit]

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 from 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.

• If both functions are non-negative, the product function ${\displaystyle fg}$ of two lower semicontinuous functions is lower semicontinuous. The corresponding result holds for upper semicontinuous functions.

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 ${\displaystyle n=1,2,...}$ .

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.

And 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.

(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.)

See also [edit]

• Directional continuity
• Katětov–Tong insertion theorem – On existence of a continuous function between semicontinuous upper and lower bounds
• Semicontinuous multivalued function

Notes [edit]

References [edit]

1. ^ Verry, Matthieu. "Histoire des mathématiques - René Baire".
2. ^ ^{a b} Stromberg, p. 132, Exercise 4
3. ^ Willard, p. 49, problem 7K
4. ^ Giaquinta, Mariano (2007). Mathematical analysis : linear and metric structures and continuity. Giuseppe Modica (1 ed.). Boston: Birkhäuser. Theorem 11.3, p.396. ISBN978-0-8176-4514-4. OCLC 213079540.
5. ^ Puterman, Martin L. (2005). Markov Decision Processes Discrete Stochastic Dynamic Programming . Wiley-Interscience. pp. 602. ISBN978-0-471-72782-8.
6. ^ Moore, James C. (1999). Mathematical methods for economic theory . Berlin: Springer. p. 143. ISBN9783540662358.
7. ^ "To show that the supremum of any collection of lower semicontinuous functions is lower semicontinuous".
8. ^ Stromberg, p. 132, Exercise 4(g)
9. ^ "Show that lower semicontinuous function is the supremum of an increasing sequence of continuous functions".

Bibliography [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. ISBN0-201-00636-7.
• Bourbaki, Nicolas (1998). Elements of Mathematics: General Topology, 5–10. Springer. ISBN3-540-64563-2.
• Engelking, Ryszard (1989). General Topology. Heldermann Verlag, Berlin. ISBN3-88538-006-4.
• Gelbaum, Bernard R.; Olmsted, John M.H. (2003). Counterexamples in analysis. Dover Publications. ISBN0-486-42875-3.
• Hyers, Donald H.; Isac, George; Rassias, Themistocles M. (1997). Topics in nonlinear analysis & applications. World Scientific. ISBN981-02-2534-2.
• Stromberg, Karl (1981). Introduction to Classical Real Analysis. Wadsworth. ISBN978-0-534-98012-2.
• Willard, Stephen (2004) [1970]. General Topology (First ed.). Mineola, N.Y.: Dover Publications. ISBN978-0-486-43479-7. OCLC 115240.
• Zălinescu, Constantin (30 July 2002). Convex Analysis in General Vector Spaces . River Edge, N.J. London: World Scientific Publishing. ISBN978-981-4488-15-0. MR 1921556. OCLC 285163112 – via Internet Archive.

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Source: https://en.wikipedia.org/wiki/Semi-continuity

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