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 f {\displaystyle f} is upper (respectively, lower) semicontinuous at a point x 0 {\displaystyle x_{0}} if, roughly speaking, the function values for arguments near x 0 {\displaystyle x_{0}} are not much higher (respectively, lower) than f ( x 0 ) . {\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 x 0 {\displaystyle x_{0}} to f ( x 0 ) + c {\displaystyle f\left(x_{0}\right)+c} for some c > 0 {\displaystyle c>0} , then the result is upper semicontinuous; if we decrease its value to f ( x 0 ) c {\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 f ( x 0 ) . {\displaystyle f\left(x_{0}\right).}

A lower semicontinuous function that is not upper semicontinuous. The solid blue dot indicates f ( x 0 ) . {\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 X {\displaystyle X} is a topological space and f : X R ¯ {\displaystyle f:X\to {\overline {\mathbb {R} }}} is a function with values in the extended real numbers R ¯ = R { , } = [ , ] {\displaystyle {\overline {\mathbb {R} }}=\mathbb {R} \cup \{-\infty ,\infty \}=[-\infty ,\infty ]} .

Upper semicontinuity [edit]

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

lim sup x x 0 f ( x ) f ( x 0 ) {\displaystyle \limsup _{x\to x_{0}}f(x)\leq f(x_{0})}

where lim sup is the limit superior of the function f {\displaystyle f} at the point x 0 {\displaystyle x_{0}} .

A function f : X R ¯ {\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 f 1 ( ( , y ) ) = { x X : f ( x ) < y } {\displaystyle f^{-1}((\leftarrow ,y))=\{x\in X:f(x)<y\}} with y R {\displaystyle y\in \mathbb {R} } are open in X {\displaystyle X} , where ( , y ) = { t R ¯ : t < y } {\displaystyle (\leftarrow ,y)=\{t\in {\overline {\mathbb {R} }}:t<y\}} .
(3) All superlevel sets { x X : f ( x ) y } {\displaystyle \{x\in X:f(x)\geq y\}} with y R {\displaystyle y\in \mathbb {R} } are closed in X {\displaystyle X} .
(4) The hypograph { ( x , t ) X × R : t f ( x ) } {\displaystyle \{(x,t)\in X\times \mathbb {R} :t\leq f(x)\}} is closed in X × R {\displaystyle X\times \mathbb {R} } .
(5) The function is continuous when the codomain R ¯ {\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 ( , y ) {\displaystyle (\leftarrow ,y)} .

Lower semicontinuity [edit]

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

lim inf x x 0 f ( x ) f ( x 0 ) {\displaystyle \liminf _{x\to x_{0}}f(x)\geq f(x_{0})}

where lim inf {\displaystyle \liminf } is the limit inferior of the function f {\displaystyle f} at point x 0 {\displaystyle x_{0}} .

A function f : X R ¯ {\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 f 1 ( ( y , ) ) = { x X : f ( x ) > y } {\displaystyle f^{-1}((y,\rightarrow ))=\{x\in X:f(x)>y\}} with y R {\displaystyle y\in \mathbb {R} } are open in X {\displaystyle X} , where ( y , ) = { t R ¯ : t > y } {\displaystyle (y,\rightarrow )=\{t\in {\overline {\mathbb {R} }}:t>y\}} .
(3) All sublevel sets { x X : f ( x ) y } {\displaystyle \{x\in X:f(x)\leq y\}} with y R {\displaystyle y\in \mathbb {R} } are closed in X {\displaystyle X} .
(4) The epigraph { ( x , t ) X × R : t f ( x ) } {\displaystyle \{(x,t)\in X\times \mathbb {R} :t\geq f(x)\}} is closed in X × R {\displaystyle X\times \mathbb {R} } .
(5) The function is continuous when the codomain R ¯ {\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 ( y , ) {\displaystyle (y,\rightarrow )} .

Examples [edit]

Consider the function f , {\displaystyle f,} piecewise defined by:

f ( x ) = { 1 if x < 0 , 1 if x 0 {\displaystyle f(x)={\begin{cases}-1&{\mbox{if }}x<0,\\1&{\mbox{if }}x\geq 0\end{cases}}}

This function is upper semicontinuous at x 0 = 0 , {\displaystyle x_{0}=0,} but not lower semicontinuous.

The floor function f ( x ) = x , {\displaystyle f(x)=\lfloor x\rfloor ,} which returns the greatest integer less than or equal to a given real number x , {\displaystyle x,} is everywhere upper semicontinuous. Similarly, the ceiling function f ( x ) = x {\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

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

is upper semicontinuous at x = 0 {\displaystyle x=0} while the function limits from the left or right at zero do not even exist.

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

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

Properties [edit]

Unless specified otherwise, all functions below are from a topological space X {\displaystyle X} to the extended real numbers R ¯ = [ , ] {\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 f : X R ¯ {\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 f g {\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 f 1 f 2 f 3 {\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 f n ( x ) = 1 ( 1 x ) n {\displaystyle f_{n}(x)=1-(1-x)^{n}} defined for x [ 0 , 1 ] {\displaystyle x\in [0,1]} for n = 1 , 2 , . . . {\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 f : X R ¯ {\displaystyle f:X\to {\overline {\mathbb {R} }}} is the limit of a monotone decreasing sequence of extended real-valued continuous functions on X {\displaystyle X} ; if f {\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, f {\displaystyle f} is continuous when R ¯ {\displaystyle {\overline {\mathbb {R} }}} is given the left order topology. So its image f ( C ) {\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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