A Unified Approach to Extended Real-Valued Functions
aa r X i v : . [ m a t h . O C ] J un A UNIFIED APPROACH TO EXTENDED REAL-VALUEDFUNCTIONS byPETRA WEIDNER Research ReportVersion 6 from June 8, 2018Extended Version of Version 1 from November 16, 2015
Abstract.
Extended real-valued functions are often used in optimization theory, but indifferent ways for infimum problems and for supremum problems. We present an approachto extended real-valued functions that works for all types of problems and into whichresults of convex analysis can be embedded. Our approach preserves continuity and theChebyshev norm when extending a functional to the entire space. The basic idea also worksfor other spaces than R . Moreover, we illustrate that extended real-valued functions haveto be handled in another way than real-valued functions and characterize semicontinuity,convexity, linearity and related properties of such functions. Keywords:
Extended real-valued function; Semicontinuity; Convex function; Linearfunction; Sublevel set; Epigraph; Indicator function
Mathematics Subject Classification (2010): HAWK Hochschule Hildesheim/Holzminden/G¨ottingen, University of Applied Sciencesand Arts, Faculty of Natural Sciences and Technology, D-37085 G¨ottingen, Germany,[email protected]. Introduction
In this report, we present basic notations and properties for functions whichattain values in R := R ∪ {−∞ , + ∞} , the extended set of real numbers . Ofcourse, such a function may also be real-valued, but infima, suprema, limits, im-proper integrals or some measure can result in function values −∞ or + ∞ . Suchvalues can also result from the extension of a function to the entire space or from theaddition of an indicator function expressing certain restrictions in convex analysis.Our approach differs from the usual one in convex analysis presented in [12]and [13], but results from convex analysis can be embedded into our approach aswe will show in Section 8. The main difference is the replacement of + ∞ if it isused as a symbol for infeasibility by a symbol which does not belong to R . Thisresults in a unified calculus for extended real-valued functions that does not dependon the convexity of the function or on whether the function is to be minimizedor maximized. The necessity for an alternative approach to extended real-valuedfunctions came up when studying properties of the function ϕ A,k given by(1.1) ϕ A,k ( x ) := inf { t ∈ R | x ∈ tk + A } for some set A ⊂ X and some k ∈ X \ { } in a linear space X under assumptionswhich do not guarantee that the functional is defined and real-valued on the entirespace. Depending on the set A and the vector k , ϕ A,k may be convex, concave orneither. Moreover, ϕ A,k may be of interest in minimization and in maximizationproblems. If the function is convex (e.g. for X = R , A = { ( x , x ) T | x ≤ , x ≤ } , k = (1 , T ), it is of importance for vector minimization problems. If itis concave (e.g. for X = R , A = { ( x , x ) T | ≤ x ≤ e x } , k = ( − , T ), it canbe used in economics as a utility function which has to be maximized.Let us underline that it was not our aim to change the approach in convexanalysis, but that we wanted to find a way for extending functions to the entirespace in order to use the tools of convex analysis without fixing in advance whetherthe function would be minimized or maximized. One can easily transfer resultsfrom convex analysis, especially from variational analysis, to functions which aregiven according to our approach.Beside the unified calculus, our extension of functions preserves continuity, semi-continuity, suprema and infima of functionals. We get rid of the irritating inequalitysup ∅ < inf ∅ and can, e.g., apply the Chebyshev norm approximation immediatelyto extended real-valued functions which have been extended to the entire spacesince, in our calculus, the expressioninf y ∈ Y sup x ∈ X | f ( x ) − g ( x, y ) | makes sense for arbitrary extended real-valued functions f , g defined on spaces X and X × Y .In the following two sections, we will give a short overview about the extended setof real numbers and about the way extended real-valued functionals are handled inconvex analysis. Here we will also discuss problems which arise by admitting values ±∞ and point out more in detail why we work with a unified approach to extendedreal-valued functions instead of the standard approach of convex analysis. Theunified approach will be described in Section 4. Section 5 is devoted to continuityand semicontinuity, whereas Section 6 deals with convexity of extended real-valuedfunctions. In both sections, we also prove basic facts about semicontinuity and convexity which have to be adapted to our framework. Well known results usingthe epigraph or sublevel sets will appear in a new light. In our proofs, we have totake into consideration that + ∞ can belong to the effective domain of the function,that we admit a symbolic function value ν / ∈ R and that results of other authorsare sometimes based on wrong rules for the calculation in R . Since in our theoryconvexity and concavity have not to be treated separately, affinity and linearityof functions which are not finite-valued can be investigated. This will be donein Section 7. Section 8 shows in which way results from convex analysis can betransferred to the unified approach and vice versa and that there exists a one-to-one correspondence between extended real-valued functions in convex analysis anda subset of the extended real-valued functions in the unified approach. In the lastsection, we will point out that the unified approach can easily be extended to otherspaces than R .From now on, R , Z , N and Q will denote the sets of real numbers, of integers, ofnonnegative integers and of rational numbers, respectively. We define R + := { x ∈ R | x ≥ } . Linear spaces will always be assumed to be real vector spaces. A set C in a linear space is a cone if λc ∈ C for all λ ∈ R + , c ∈ C . For a subset A of somelinear space, icr( A ) will denote the relative algebraic interior of A . In a topologicalspace X , N ( x ) is the set of all neighborhoods of x ∈ X and cl A , int A and bd A denote the closure, the interior and the boundary, respectively, of a subset A .2. The extended set of real numbers
The extended set of real numbers, R = R ∪ {−∞ , + ∞} , is especially used inmeasure theory and in convex analysis. In measure theory, a set may have themeasure + ∞ . In convex analysis, the following properties of R are of interest: • ( R , ≤ ) is a totally ordered set and each subset of R has an infimum and asupremum, • R is a compact Hausdorff space.Let us first recall that in R the following rules are defined for the calculationwith ±∞ . ∀ y ∈ R : −∞ < y < + ∞ , − (+ ∞ ) = −∞ , − ( −∞ ) = + ∞ , (+ ∞ ) · (+ ∞ ) = ( −∞ ) · ( −∞ ) = + ∞ , (+ ∞ ) · ( −∞ ) = ( −∞ ) · (+ ∞ ) = −∞ , ∀ y ∈ R with y > y · (+ ∞ ) = (+ ∞ ) · y = + ∞ ,y · ( −∞ ) = ( −∞ ) · y = −∞ , ∀ y ∈ R with y < y · (+ ∞ ) = (+ ∞ ) · y = −∞ ,y · ( −∞ ) = ( −∞ ) · y = + ∞ , ∀ y ∈ R ∪ { + ∞} : y + (+ ∞ ) = + ∞ + y = + ∞ , ∀ y ∈ R ∪ {−∞} : y + ( −∞ ) = −∞ + y = −∞ . Moreover, we define0 · (+ ∞ ) = (+ ∞ ) · · ( −∞ ) = ( −∞ ) · . R is not a real linear space since + ∞ and −∞ do not have an inverse elementw.r.t. (with regard to) addition and since the calculation within R is not possiblein an analogous way as in R . Example 1.
The rule ( λ + µ ) x = λx + µx does not hold for arbitrary λ, µ ∈ R and x ∈ { + ∞ , −∞} since otherwise, e.g., + ∞ = (3 − · (+ ∞ ) = 3 · (+ ∞ ) + ( − · (+ ∞ ) = + ∞ + ( −∞ ) = 2 · (+ ∞ ) + ( − · (+ ∞ ) = (2 − · (+ ∞ ) = −∞ . Though R is totally ordered, equivalent transformations of equations and in-equalities in R do not work in the same way in R . Example 2. In R , a + b = a + c does not imply b = c , and a + b ≤ a + c does notimply b ≤ c . Choose, e.g., a = −∞ , b = 0 and c = − . But we can equip R with the topology that is generated by a neighborhood baseof the Euclidean topology on R together with the neighborhood bases B (+ ∞ ) = {{ x ∈ R | x > a } | a ∈ R } and B ( −∞ ) = {{ x ∈ R | x < a } | a ∈ R } . Wheneverwe will consider R as a topological space, this topology will be supposed. Let usnote that R is a compact Hausdorff topological space, but not a topological vectorspace. Nevertheless, the following rules are valid in its topology τ :(a) ∀ O ∈ τ \ {∅} ∀ λ ∈ R \ { } : λO ∈ τ .(b) ∀ O ∈ τ \ {∅} ∀ x ∈ R : x + O ∈ τ .3. Extended real-valued functions in convex analysis
The approach in this section is part of the basics of convex analysis as theywere presented by Moreau [12] and Rockafellar [13] and has become standard whenstudying minimization problems (see, e.g., [1], [8], [2], [16], [3]), which includesthe variational analysis for such problems ([14], [10], [11]). Originally the conceptfocused on the minimization of real-valued convex functions which are extended tothe complete space by the value + ∞ and/or for which the barrier + ∞ indicatesthe violation of restrictions. When minimizing a function in the case that feasiblesolutions exist, the function value + ∞ does not alter the minimum.The consideration of extended real-valued functions in convex analysis is unilat-eral. The handling of the function depends on whether it is to be minimized or tobe maximized. The usual definitions are given for minimization problems and canbe applied to maximization by working with −∞ instead of + ∞ where the rulefor the addition of + ∞ and −∞ has to be changed. The definition of an indicatorfunction also refers to the minimization of the function to which it is added.We will now explain the usual approach for a function f that has to be minimized.Even if f would be real-valued it could be extended to a function that is not real-valued any more by the following two reasons. f : C → R with C being a subset of some space X can be extended to a function g : X → R by defining g ( x ) := (cid:26) f ( x ) if x ∈ C, + ∞ if x ∈ X \ C. Functions with values in R also play an important role in optimization theory ifthe problem inf { f ( x ) | x ∈ C } with f : X → R is replaced by inf { g ( x ) | x ∈ X } with g := f + Ψ C , where Ψ C : X → R is the indicator function of C defined byΨ C ( x ) := (cid:26) x ∈ C, + ∞ if x ∈ X \ C. Here, the rule + ∞ + ( −∞ ) = + ∞ has to be applied.Working with g instead of f offers an elegant way to avoid the necessity forinvestigating technical details of C in advance. E.g. when studying the functional ϕ A,k given by equation (1.1) one can simply write ϕ A,k : X → R using the definitioninf ∅ = + ∞ .Obviously, the value + ∞ serves as a symbol for a function value at points whichare not feasible for the problem, and if a function value + ∞ exists since, e.g., thefunction is defined via a supremum, the related argument of the function is handledas if it would not belong to the domain of interest.Consequently, the (effective) domain of a function g : X → R is defined as { x ∈ X | g ( x ) ∈ R ∪ {−∞}} .Based on this classical approach, useful tools for convex optimization have beendeveloped, but we have to point out that extended real-valued functions have tobe handled carefully. One has to take into consideration that the familiar laws ofarithmetic are not valid in full extent on R (cp. Example 1) and that equationsas well as inequalities cannot be handled in the same way as for real numbers (cp.Example 2). Let us e.g. recall that a function f : C → R , C being a convex subsetof a linear space X , is said to be convex on C if f ( λx + (1 − λ ) x ) ≤ λf ( x ) + (1 − λ ) f ( x )holds for all x , x ∈ C and λ ∈ (0 , R hasto be defined in an alternative way, e.g. via convexity of the epigraph of f (cp. e.g.[13]).We will now point out drawbacks of the approach in convex analysis which willnot appear in the unified approach we are going to present in the next section.(a) The way an extended real-valued function in convex analysis is handleddepends on whether it has to be minimized or to be maximized. Supremumproblems are studied in an analogous way as the infimum problems above,but with −∞ replacing + ∞ and vice versa. One consequence is the differentdefinition of + ∞ + ( −∞ ) in both frameworks. The so-called inf-addition+ ∞ + ( −∞ ) = ( −∞ ) + (+ ∞ ) = + ∞ is applied for minimization problems, whereas the sup-addition for maxi-mization problems is given by+ ∞ + ( −∞ ) = ( −∞ ) + (+ ∞ ) = −∞ . Moreau [12] introduced both kinds of addition and, in connection with this,two different addition operators on R . Note the following consequence ofthe inf-addition for a, b, c ∈ R : a ≤ b + c implies − a ≥ − ( b + c ), but not − a ≥ − b − c since, e.g., 3 ≤ (+ ∞ ) + ( −∞ ), but − ( −∞ ) + (+ ∞ ). Nevertheless, a ≤ b + c = ⇒ − a ≥ − b − c if {−∞ , + ∞} 6 = { b, c } . (b) The extension of a convex function f : C → R , C being a proper convexsubset of some topological vector space X , to g : X → R by adding thefunction value + ∞ outside C can destroy the continuity and even the lowersemicontinuity of f . Example 3.
Consider the functional ϕ : R → R given by ϕ ( x ) = (cid:26) x if x > , + ∞ if x ≤ .ϕ is continuous on { x ∈ R | ϕ ( x ) ∈ R ∪ {−∞}} , but not lower semiconti-nuous on R . (c) Multiplication of functions with real numbers and subtraction of functionsare only possible under certain assumptions [8, p.389].(d) In connection with extended real-valued functions in convex analysis, thedefinitions inf ∅ = + ∞ and sup ∅ = −∞ are given. Hence sup ∅ < inf ∅ .(e) If a function is extended to the entire space by + ∞ or −∞ it cannot behandled like the original function any more. This is obviously the casesince the way of extension depends on the purpose. The extended func-tion does also not have a finite Chebyshev norm and the Chebyshev normapproximation inf y ∈ Y sup x ∈ X | f ( x ) − g ( x, y ) | cannot be applied.(f) In some optimization problems a function cannot simply be extended tothe entire space using only one of the values + ∞ or −∞ . Consider theproblem(3.1) inf x ∈ X sup y ∈ Y L ( x, y ) , where Y is a proper subset of some space Y , X is a proper subset of somespace X and L is real-valued. In such a case L is extended to a function ℓ which is defined on X × Y and for which the probleminf x ∈ X sup y ∈ Y ℓ ( x, y )is equivalent to problem (3.1) in the following way [14, Example 11.52]: ℓ ( x, y ) = L ( x, y ) if x ∈ X , y ∈ Y , −∞ if x ∈ X , y ∈ Y \ Y , + ∞ if x ∈ X \ X . (g) The previous item illustrates that the unilateral approach in convex analysisbecomes complicated when infimum problems and supremum problems arecombined. Duality theory usually works with such combined problems.Then one has to be very careful in using the notion of the effective domainor adding + ∞ and −∞ . The usual indicator function in convex analysisshould not be added to functions which have to be maximized. Unified approach to extended real-valued functions
We introduce the symbol ν as a function value in arguments which are notfeasible otherwise. Keeping this in mind, definitions of notions and properties forextended real-valued functions will emerge in a natural way. For sets A ⊆ R wedefine A ν := A ∪ { ν } .On subsets of a space X ( X = ∅ ), we consider functions which can take values in R . If a function ϕ is defined on a subset X ⊆ X , we extend the range of definitionto the entire space X by defining ϕ ( x ) := ν for all x ∈ X \ X . This yields afunction ϕ : X → R ν . We call functions with values in R ν extended real-valuedfunctions and often refer to them simply as functionals. Definition 1.
Consider an extended real-valued function ϕ : X → R ν on somenonempty set X . We define its (effective) domain as dom ϕ := { x ∈ X | ϕ ( x ) ∈ R } .ϕ is trivial if dom ϕ = ∅ . ϕ is called finite-valued on X ⊆ X if ϕ ( x ) ∈ R for all x ∈ X . ϕ is said to be finite-valued if ϕ is finite-valued on X . ϕ is called proper if ϕ is nontrivial and finite-valued on dom ϕ . Otherwise, it issaid to be improper .For ϕ being nontrivial, we introduce ϕ eff : dom ϕ → R by ϕ eff ( x ) = ϕ ( x ) for all x ∈ dom ϕ and call it the effective part of ϕ . Remark 1.
Our definition of the effective domain essentially differs from the usualdefinition in convex analysis (see Section 3) since we use ν instead of + ∞ andadmit + ∞ to be a function value that comes into existence, e.g., by defining thefunction as some supremum, and may be of interest in problems which depend onthe function. The finite-valued functionals are just the functions ϕ : X → R and are just theproper functionals with dom ϕ = X . For finite-valued functionals, ϕ eff = ϕ . Theproper functions are the nontrivial functions ϕ : X → R ν .Each function and operation applied to ν has to result in ν if the function mapsinto R ν , and to result in the empty set if the function value should be a set.In R ν , the following rules are defined for the calculation with ν : ∀ y ∈ R : ν y, y ν, − ν = ν, + ∞ + ( −∞ ) = −∞ + (+ ∞ ) = ν, ∀ y ∈ R ν : y · ν = ν · y = ν, ∀ y ∈ R ν : y + ν = ν + y = ν. The above definitions extend the binary relations < , ≤ , > , ≥ , =, the binaryoperations + and · as well as the unary operation − to R ν . Note that the relations < , , > and coincide with ≥ , > , ≤ and < , respectively, on R , but not on R ν .Take also into consideration that we have defined − y for all y ∈ R ν as a unaryoperation, but that y + ( − y ) = ν = 0 for y ∈ {−∞ , + ∞ , ν } . With this unaryoperation, we can define the subtraction on R ν by y − y := y + ( − y ) for all y , y ∈ R ν . Moreover, we define | + ∞| = | − ∞| = + ∞ and | ν | = ν. Now we can transfer these definitions to functions.For some nonempty set X , functions f, g : X → R ν and λ ∈ R , we define − f : X → R ν by ( − f )( x ) = − f ( x ) for all x ∈ X,λ · f : X → R ν by ( λ · f )( x ) = λ · f ( x ) for all x ∈ X,f + g : X → R ν by ( f + g )( x ) = f ( x ) + g ( x ) for all x ∈ X,f + λ : X → R ν by ( f + λ )( x ) = f ( x ) + λ for all x ∈ X,λ + f : X → R ν by λ + f = f + λ. Note thatdom( f + g ) = (dom f ∩ dom g ) \ { x ∈ X | f ( x ) ∈ {−∞ , + ∞} and g ( x ) = − f ( x ) } , dom( f + λ ) = dom( λ + f ) = dom f \ { x ∈ X | f ( x ) = − λ } if λ ∈ {−∞ , + ∞} . Extended real-valued functions can be used as indicator functions of sets.
Definition 2.
For a subset A of a space X , the indicator function of A is thefunction ι A : X → R ν defined by ι A ( x ) := (cid:26) if x ∈ A,ν if x ∈ X \ A. Remark 2.
The usual definition of indicator functions in convex analysis uses + ∞ instead of ν . The indicator functions in measure theory are defined with values and instead of and ν , respectively. The extension of functions to the whole space is not the only reason for theimportance of extended real-valued functions. Such functions also yield the possi-bility to replace an optimization problem with side conditions by a free optimizationproblem. If we are looking for optimal values of some function f : X → R ν on theset A ⊂ X , then this problem is equivalent to the calculation of optimal values ofthe function g : X → R ν if g := f + ι A .Since infima and suprema play a central role in optimization theory, we nowextend these notions to R ν . Definition 3.
A set A ⊆ R ν is called trivial if A ⊆ { ν } . b ∈ R is a lower bound of a nontrivial set A ⊆ R ν if a < b for all a ∈ A . b ∈ R is an upper bound of anontrivial set A ⊆ R ν if a > b for all a ∈ A . A nontrivial set A ⊆ R ν is boundedbelow or bounded above if there exists some real lower bound or some real upperbound, respectively, of A . It is bounded if it is bounded below and bounded above.The infimum inf A and the supremum sup A of A ⊆ R ν are defined by inf A := (cid:26) ν if A is trivial , the largest lower bound of A otherwise , sup A := (cid:26) ν if A is trivial , the smallest upper bound of A otherwise . We say that A has a minimum min A if inf A ∈ A ∩ R and define, in this case, min A := inf A . We say that A has a maximum max A if sup A ∈ A ∩ R anddefine, in this case, max A := sup A . Obviously, −∞ is a lower bound and + ∞ is an upper bound of each nontrivialset in R ν . If A ∩ R = { + ∞} , then + ∞ is also a lower bound of A . We getinf ∅ = sup ∅ = ν . For nonempty sets in R , the above notions coincide with theusual ones.Now we can define the Minkowski functional in our framework. Definition 4.
Consider some subset A of a linear space X with ∈ A . The Minkowski functional p A : X → R ν of A is defined by p A ( x ) = inf { λ > | x ∈ λA } . Then the Minkowski functional of a cone is just its indicator function [12].Let us adapt notions which are needed for dealing with differentiability and otherproperties of functions to extended real-valued functions.
Definition 5.
Assume that X is an arbitrary set and ϕ : X → R ν .The infimum and the supremum of ϕ (on X ) are defined by inf x ∈ X ϕ ( x ) :=inf { ϕ ( x ) | x ∈ X } and sup x ∈ X ϕ ( x ) := sup { ϕ ( x ) | x ∈ X } , respectively.If { ϕ ( x ) | x ∈ X } has a minimum or maximum, we say that ϕ attains a (global)minimum or a (global) maximum , respectively, on X and denote it by min x ∈ X ϕ ( x ) or max x ∈ X ϕ ( x ) , respectively.Suppose now dom ϕ = ∅ . t ∈ R is called an upper bound or a lower bound of ϕ (on X ) if it is an upper orlower bound of { ϕ ( x ) | x ∈ X } , respectively. ϕ is called bounded above , boundedbelow or bounded (on X ) if { ϕ ( x ) | x ∈ X } is bounded above, bounded below orbounded, respectively. Take into consideration that each nontrivial extended real-valued functional hasan upper bound and a lower bound, but that it is only bounded above or boundedbelow if there exists some real upper bound or lower bound, respectively. Forfunctions with values in R only, the above definition and the next one are compatiblewith the usual notions. Definition 6.
Suppose that X is a topological space and ϕ : X → R ν .If x ∈ cl dom ϕ and if there exists some g ∈ R such that for each neighborhood V of g there exists some neighborhood U of x with ϕ ( x ) ∈ V for all x ∈ U ∩ dom ϕ ,then g is said to be the limit of ϕ at x . If x ∈ X \ cl dom ϕ or if there does notexist some limit of ϕ at x in R , then the limit of ϕ at x is defined as ν . The limit of ϕ at x is denoted by lim x → x ϕ ( x ) . For derivatives of functions, we also need limits of sequences.
Definition 7.
A function mapping N into R is called an (extended real-valued)sequence . We will denote such a sequence by ( a n ) , where a n is the function valueof n for each n ∈ N . The sequence is bounded if the mapping is bounded. A value g ∈ R is said to be a limit of ( a n ) , denoted as lim n → + ∞ a n , if for each neighborhood U of g there exists some n ∈ N such that a n ∈ U for each n ∈ N with n > n . Ifthe sequence does not have any limit in R , then ν is defined to be the limit of thesequence.A value g ∈ R is a cluster point of ( a n ) if each neighborhood of g contains a n for an infinite number of elements n ∈ N . Let C denote the set of cluster pointsof ( a n ) . sup C is called the limit superior lim sup a n , inf C is called the limitinferior lim inf a n of ( a n ) . For real-valued sequences, the above notions work as usual.Example 1 demonstrates that linear combinations of improper functionals haveto be used carefully. This requires alternative definitions of important propertieslike convexity of functions. The alternative way of describing such a propertysometimes uses the epigraph or the hypograph of a function. Definition 8.
Assume that X is a nonempty set and ϕ : X → R ν . The epigraph of ϕ is defined by epi ϕ := { ( x, t ) ∈ X × R | ϕ ( x ) ≤ t } . The hypograph or subgraph of ϕ is the set hypo ϕ := { ( x, t ) ∈ X × R | ϕ ( x ) ≥ t } . Whenever X is a topological space, the investigation of topological properties ofthe epigraph or of the hypograph refers to the space X × R .Let us mention some immediate consequences of the definition. Lemma 1.
Let X be a nonempty set and ϕ : X → R ν . (a) ( x, t ) ∈ epi ϕ if and only if ( x, − t ) ∈ hypo( − ϕ ) . (b) For each x ∈ X , ϕ ( x ) = −∞ if and only if { x } × R ⊆ epi ϕ . In this case, ( { x } × R ) ∩ hypo ϕ = ∅ . (c) For each x ∈ X , ϕ ( x ) = + ∞ if and only if { x } × R ⊆ hypo ϕ . In this case, ( { x } × R ) ∩ epi ϕ = ∅ . Part (a) of this lemma yields:
Corollary 1.
Let X be a linear space and ϕ : X → R ν . (a) hypo ϕ + hypo ϕ ⊆ hypo ϕ ⇐⇒ epi( − ϕ ) + epi( − ϕ ) ⊆ epi( − ϕ ) . (b) hypo ϕ is convex ⇐⇒ epi( − ϕ ) is convex. (c) hypo ϕ is a cone ⇐⇒ epi( − ϕ ) is a cone. Corollary 2.
Let X be a topological space and ϕ : X → R ν . Then hypo ϕ is closed ⇐⇒ epi( − ϕ ) is closed. Continuity and Semicontinuity
We now extend the definition of continuity to functionals ϕ : X → R ν . Definition 9.
Let X be a topological space, ϕ : X → R ν . ϕ is continuous at x ∈ X if x ∈ dom ϕ and ϕ eff is continuous at x . ϕ is continuous on the nonemptyset D ⊆ X if D ⊆ dom ϕ and ϕ eff is continuous on D . If dom ϕ is nonempty andclosed, then we call ϕ a continuous functional if ϕ is continuous on dom ϕ . Example 4.
Consider the functional ϕ : R → R ν given by ϕ ( x ) = tan( x ) if − π < x < π , −∞ if x = − π , + ∞ if x = π ,ν if x < − π or x > π . dom ϕ = { x ∈ R | − π ≤ x ≤ + π } is closed, and ϕ is a continuous functional. Note that the previous example also illustrates that there exists a one-to-one-correspondence between the closed interval [ − π , π ] and R and also a one-to-one-correspondence between the open sets induced on [ − π , π ] by the Euclidean topologyand the open sets in R . Indeed, it is well known that R can be considered as atwo-point compactification of R since (as previously mentioned) R is a compacttopological space.Functionals which are continuous on dom ϕ can often be extended in such a waythat the domain of the extended functional is closed and the extended functionalis continuous. Example 5.
Let ϕ : R → R ν be given by ϕ ( x ) = (cid:26) | x | if x = 0 ,ν if x = 0 .ϕ is continuous on dom ϕ , but dom ϕ is not closed. ϕ can be extended to a continu-ous functional with dom ϕ = R by replacing the symbolic function value ν in x = 0 by + ∞ . Of course, not each functional which is continuous on its effective domain canbe extended to a continuous function, e.g., if there exists a jump between two partsof the effective domain. But in contrast to the usual approach in convex analysis,semicontinuity and continuity of a functional cannot be destroyed by extending thefunction to the entire space using ν .The definition of continuity implies: Lemma 2.
Let X be a topological space and ϕ : X → R ν . (a) If ϕ is continuous at x ∈ dom ϕ , then the functions λϕ and ϕ + λ with λ ∈ R are continuous at x . (b) If X is a topological vector space, x , x ∈ X , x − x ∈ dom ϕ and ϕ iscontinuous at x − x , then g : X → R ν defined by g ( x ) = ϕ ( x − x ) iscontinuous at x . (c) If X is a topological vector space, x ∈ X , λ ∈ R , λx ∈ dom ϕ and ϕ iscontinuous at λx , then g : X → R ν defined by g ( x ) = ϕ ( λx ) is continuousat x . The product of continuous extended real-valued functions is not necessarily con-tinuous.
Example 6.
Define f, g : R → R by f ( x ) = (cid:26) | x | if x = 0 , + ∞ if x = 0 ,g ( x ) := | x | . Then h ( x ) := g ( x ) · f ( x ) = (cid:26) if x = 0 , · (+ ∞ ) = 0 if x = 0 . dom f = dom g = R . f and g are continuous functions, but h is not continuous at x = 0 . Let us now introduce semicontinuity of functionals, first for functionals withvalues in R only. Definition 10.
Consider a topological space X , a nonempty set D ⊆ X and afunction ϕ : D → R . ϕ is called lower semicontinuous at x ∈ D if ϕ ( x ) = −∞ or for each h ∈ R with h < ϕ ( x ) there exists some U ∈ N ( x ) such that ϕ ( x ) > h for all x ∈ U ∩ D . ϕ is called upper semicontinuous at x ∈ D if ϕ ( x ) = + ∞ or for each h ∈ R with h > ϕ ( x ) there exists some U ∈ N ( x ) such that ϕ ( x ) < h for all x ∈ U ∩ D . ϕ is called a lower semicontinuous function or an upper semicontinuousfunction (on D ) if ϕ is lower semicontinuous or upper semicontinuous, respec-tively, at each x ∈ D . We now extend this definition to functionals with values in R ν . Definition 11.
Let X be a topological space, ϕ : X → R ν . ϕ is lower semicontinuous or upper semicontinuous at x ∈ X if x ∈ dom ϕ and ϕ eff is lower semicontinuous or upper semicontinuous, respectively, at x . ϕ is lower semicontinuous or upper semicontinuous on the nonempty set D ⊆ X if D ⊆ dom ϕ and ϕ eff is lower semicontinuous or upper semicontinuous, respec-tively, on D . If dom ϕ is nonempty and closed, then we call ϕ a lower semicon-tinuous function or an upper semicontinuous function if ϕ is lower semi-continuous or upper semicontinuous, respectively, on dom ϕ . Note that each functional which can attain only one value c ∈ R on dom ϕ islower semicontinuous, upper semicontinuous and continuous on dom ϕ .Immediately from the definitions we get the following statements. Proposition 1.
Let X be a topological space, ϕ : X → R ν , x ∈ dom ϕ . (a) ϕ is upper semicontinuous at x iff − ϕ is lower semicontinuous at x . (b) ϕ is continuous at x iff ϕ is lower semicontinuous and upper semicontinuousat x . Definition 11 implies:
Lemma 3.
Let X be a topological space and ϕ : X → R ν . (a) If ϕ is lower semicontinuous at x ∈ dom ϕ , then the functions λϕ with λ ∈ R + and ϕ + c with c ∈ R are lower semicontinuous at x . (b) If X is a topological vector space, x , x ∈ X , x − x ∈ dom ϕ and ϕ is lowersemicontinuous at x − x , then g : X → R ν defined by g ( x ) = ϕ ( x − x ) islower semicontinuous at x . (c) If X is a topological vector space, x ∈ X , λ ∈ R , λx ∈ dom ϕ and ϕ islower semicontinuous at λx , then g : X → R ν defined by g ( x ) = ϕ ( λx ) islower semicontinuous at x .The same statements hold for upper semicontinuity. Corollary 3.
Let ( X, τ ) be a topological space, ϕ : X → R ν , D ⊆ dom ϕ with D = ∅ . Then the following properties of ϕ are equivalent to each other: (a) ϕ is lower semicontinuous on D . (b) Each set { x ∈ D | ϕ ( x ) > t } , t ∈ R , is open w.r.t. the topology induced by τ on D. (c) Each set { x ∈ D | ϕ ( x ) ≤ t } , t ∈ R , is closed w.r.t. the topology induced by τ on D. The next propositions will connect semicontinuity of a functional ϕ with topo-logical properties of the sublevel sets and of the sets { x ∈ X | ϕ ( x ) > t } withoutreferring explicitly to some induced topology. Definition 12.
Let R denote some binary relation on R ν , X be a nonempty set, ϕ : X → R ν . Then we define lev ϕ,R ( t ) := { x ∈ X | ϕ ( x ) Rt } for t ∈ R . Proposition 2.
Let ϕ : X → R ν be a nontrivial function on a topological space X . (a) ϕ is lower semicontinuous on dom ϕ if the sets lev ϕ,> ( t ) are open for all t ∈ R . (b) Assume that dom ϕ is open. Then ϕ is lower semicontinuous on dom ϕ if and only if the sets lev ϕ,> ( t ) areopen for all t ∈ R . (c) If ϕ is bounded below, then:The sets lev ϕ,> ( t ) are open for all t ∈ R if and only if dom ϕ is open and ϕ is lower semicontinuous on dom ϕ .Proof. (a) results immediately from the definition of lower semicontinuity.(b) Assume that dom ϕ is open and ϕ is lower semicontinuous on dom ϕ .Propose that there exists some t ∈ R for which lev ϕ,> ( t ) is not open. ⇒ ∃ x ∈ lev ϕ,> ( t ) ∀ U ∈ N ( x ) : U lev ϕ,> ( t ). Since ϕ is lowersemicontinuous on dom ϕ , there exists some neighborhood V of x with V ⊆ lev ϕ,> ( t ) ∪ ( X \ dom ϕ ). Consider some arbitrary neighborhood V of x . V := V ∩ V ⊆ lev ϕ,> ( t ) ∪ ( X \ dom ϕ ), but V lev ϕ,> ( t ). ⇒ ∃ v ∈ V : v ∈ X \ dom ϕ. ⇒ ∀ V ∈ N ( x ) ∃ v ∈ V : v ∈ X \ dom ϕ. ⇒ x ∈ cl( X \ dom ϕ ) = X \ dom ϕ since dom ϕ is open. Thiscontradicts x ∈ lev ϕ,> ( t ) and yields the assertion.(c) Consider some lower bound c of ϕ . If lev ϕ,> ( c −
1) is open, A := X \ lev ϕ,> ( c −
1) = { x ∈ X | ϕ ( x ) ≤ c − } ∪ ( X \ dom ϕ ) = X \ dom ϕ has tobe closed and thus dom ϕ is open. This implies (c) because of (b). (cid:3) Example 4 shows a continuous functional ϕ for which not all sets lev ϕ,> ( t ) areopen.Replacing the function value ϕ ( π ) in Example 4 by ν , we get a functional ϕ which is continuous on dom ϕ = { x ∈ R | − π ≤ x < π } and for which all setslev ϕ,> ( t ), t ∈ R , are open, though dom ϕ is not open. Proposition 3.
Let ϕ : X → R ν be a nontrivial function on a topological space X . (a) ϕ is lower semicontinuous on dom ϕ if the sublevel sets lev ϕ, ≤ ( t ) are closedfor all t ∈ R . (b) Suppose that dom ϕ is closed. Then ϕ is lower semicontinuous if and only if the sublevel sets lev ϕ, ≤ ( t ) are closedfor all t ∈ R . (c) If ϕ is bounded above, then:The sublevel sets lev ϕ, ≤ ( t ) are closed for all t ∈ R if and only if dom ϕ isclosed and ϕ is lower semicontinuous on dom ϕ .Proof. (a) lev ϕ, ≤ ( t ) is closed if and only if A ( t ) := X \ lev ϕ, ≤ ( t ) = lev ϕ,> ( t ) ∪ ( X \ dom ϕ ) is open. If A ( t ) is open for all t ∈ R , the lower semicontinuity ofthe functional on dom ϕ follows from the definition of this property.(b) The definition of lower semicontinuity of ϕ implies that the set A ( t ) is openfor each t ∈ R for which lev ϕ,> ( t ) is not empty. Thus the set A ( t ) is openfor each t ∈ R if dom ϕ is closed. This yields assertion (b).(c) Consider some upper bound b of ϕ . If lev ϕ, ≤ ( b ) is closed, A := X \ lev ϕ, ≤ ( b ) = { x ∈ X | ϕ ( x ) > b } ∪ ( X \ dom ϕ ) = X \ dom ϕ has to beopen and thus dom ϕ is closed. This implies (c) because of (b). (cid:3) A function can be lower semicontinuous on its domain though not all sublevelsets lev ϕ, ≤ ( t ) are closed. Example 7.
Consider the functional ϕ : R → R ν given by ϕ ( x ) = (cid:26) x if x > ,ν if x ≤ .ϕ is continuous on dom ϕ , but lev ϕ, ≤ ( t ) is not closed if t > . In Example 5, the sublevel sets lev ϕ, ≤ ( t ) are closed for all t ∈ R , though ϕ iscontinuous on dom ϕ and dom ϕ is not closed.For upper semicontinuity, analogous statements as in the previous two propo-sitions follow with reverse relations and reverse directions of boundedness fromProposition 1.Let us now investigate the connection between lower semicontinuity and closed-ness of the epigraph. Lemma 4.
Let X be a topological space, ϕ : X → R ν . epi ϕ is closed in X × R if and only if the sublevel sets lev ϕ, ≤ ( t ) = { x ∈ X | ϕ ( x ) ≤ t } are closed for all t ∈ R .Proof. (i) epi ϕ = { ( x, t ) ∈ X × R | ϕ ( x ) ≤ t } is closed iff ( X × R ) \ epi ϕ = { ( x, t ) ∈ X × R | ϕ ( x ) > t } ∪ (( X \ dom ϕ ) × R ) is open. Then for each t ∈ R , { x ∈ X | ϕ ( x ) > t } ∪ ( X \ dom ϕ ) is open and thus { x ∈ X | ϕ ( x ) ≤ t } isclosed.(ii) Suppose that lev ϕ, ≤ ( t ) is closed for each t ∈ R . Assume that epi ϕ is notclosed. Then ( X × R ) \ epi ϕ is not open. ⇒ ∃ ( x, λ ) ∈ ( X × R ) \ epi ϕ ∀ V ∈N (( x, λ )) ∃ ( x V , t V ) ∈ V : ( x V , t V ) ∈ epi ϕ , i.e. ϕ ( x V ) ≤ t V . Consider anarbitrary ǫ > U ǫ ( λ ) := { r ∈ R | λ − ǫ < r < λ + ǫ } and an arbitrary U ∈ N ( x ). ⇒ V ǫ := U × U ǫ ( λ ) ∈ N (( x, λ )) . ⇒ ∃ ( x ǫ , t ǫ ) ∈ V ǫ : ϕ ( x ǫ ) ≤ t ǫ <λ + ǫ. ⇒ ∀ U ∈ N ( x ) ∃ x ǫ ∈ U : x ǫ ∈ lev ϕ, ≤ ( λ + ǫ ) . ⇒ x ∈ cl(lev ϕ, ≤ ( λ + ǫ )) =lev ϕ, ≤ ( λ + ǫ ) . ⇒ ϕ ( x ) ≤ λ + ǫ ∀ ǫ > . ⇒ ϕ ( x ) ≤ λ , a contradiction to( x, λ ) / ∈ epi ϕ . Thus epi ϕ is closed. (cid:3) Corollary 2 results in an analogous statement for hypo ϕ and the superlevel setslev ϕ, ≥ ( t ).Proposition 3 implies because of Lemma 4: Proposition 4.
Let ϕ : X → R ν be a nontrivial function on a topological space X . (a) ϕ is lower semicontinuous on dom ϕ if epi ϕ is closed in X × R . (b) If dom ϕ is closed, then ϕ is lower semicontinuous ⇐⇒ epi ϕ is closed in X × R . (c) If ϕ is bounded above, then: epi ϕ is closed in X × R if and only if dom ϕ is closed and ϕ is lowersemicontinuous on dom ϕ . The function ϕ in Example 7 is lower semicontinuous on its domain thoughepi( ϕ ) is not closed. In Example 5, epi ϕ is closed in X × R , though ϕ is continuouson dom ϕ and dom ϕ is not closed. For upper semicontinuity, we can prove an analogous statement as in Proposi-tion 4 by replacing the epigraph by the hypograph and changing the direction ofboundedness.Corollary 3 and Lemma 4 imply:
Corollary 4.
Let X be a topological space, ϕ : X → R .Then the following statements are equivalent: (a) ϕ is lower semicontinuous. (b) The sets lev ϕ,> ( t ) are open for all t ∈ R . (c) The sublevel sets lev ϕ, ≤ ( t ) are closed for all t ∈ R . (d) epi ϕ is closed in X × R . Remark 3.
The characterization of lower semicontinuity in Corollary 4 describesthe ways in which lower semicontinuity is defined in the classical framework whichworks with + ∞ instead of ν . Rockafellar [13, Theorem 7.1] proved the equivalencebetween these conditions for ϕ : R n → R . Moreau [12] defined lower semicontinuityby property (c) and stated the equivalence with property (d) for X being a topologicalspace. Convexity
We will now study convexity of extended real-valued functions.
Definition 13.
Let X be a linear space and ϕ : X → R ν . ϕ is said to be convex if dom ϕ and epi ϕ are convex sets. ϕ is concave if dom ϕ and hypo ϕ are convex sets. Remark 4.
The characterization of a real-valued convex function on some finite-dimensional vector space by the epigraph of the function was given by Fenchel [5,p. 57] . If ϕ : X → R ν does not attain the value + ∞ , then convexity of epi ϕ impliesconvexity of dom ϕ .A continuous functional with a convex epigraph and a convex hypograph doesnot necessarily have a convex domain. Example 8.
Define ϕ : R → R ν by ϕ ( x ) = −∞ if x = − , + ∞ if x = 1 ,ν if x ∈ R \ {− , } . Then dom ϕ = {− , } is closed, but not convex. ϕ is continuous. epi ϕ = {− }× R and hypo ϕ = { } × R are closed convex sets. Lemma 1 implies:
Lemma 5.
Let X be a linear space and ϕ : X → R ν .Then ϕ is concave on X if and only if − ϕ is convex on X . Definition 13 is compatible with the usual definition of convex real-valued func-tions. Theorem 1.
Let X be a linear space and ϕ : X → R ν be proper.Then ϕ is convex or concave if and only if dom ϕ is convex and ϕ ( λx + (1 − λ ) x ) ≤ λϕ ( x ) + (1 − λ ) ϕ ( x ) or (6.1) ϕ ( λx + (1 − λ ) x ) ≥ λϕ ( x ) + (1 − λ ) ϕ ( x ) , respectively , holds for all x , x ∈ dom ϕ , λ ∈ (0 , .Proof. (a) If ϕ is convex, then epi ϕ is a convex set. Consider arbitrary elements x , x ∈ dom ϕ , λ ∈ (0 , x , ϕ ( x )) , ( x , ϕ ( x )) ∈ epi ϕ , weget: λ · ( x , ϕ ( x ))+ (1 − λ ) · ( x , ϕ ( x )) ∈ epi ϕ . Hence ϕ ( λx + (1 − λ ) x ) ≤ λϕ ( x ) + (1 − λ ) ϕ ( x ).(b) Assume that dom ϕ is convex and ϕ ( λx +(1 − λ ) x ) ≤ λϕ ( x )+(1 − λ ) ϕ ( x )for all x , x ∈ dom ϕ , λ ∈ (0 , x , t ) , ( x , t ) ∈ epi ϕ , i.e. x , x ∈ dom ϕ with ϕ ( x ) ≤ t , ϕ ( x ) ≤ t . Then λx + (1 − λ ) x ∈ dom ϕ and ϕ ( λx + (1 − λ ) x ) ≤ λϕ ( x ) + (1 − λ ) ϕ ( x ) ≤ λt + (1 − λ ) t for all λ ∈ (0 , λ · ( x , t ) + (1 − λ ) · ( x , t ) ∈ epi ϕ for all λ ∈ (0 , ϕ is convex.Thus the assertion holds for convex functionals. This, together with Lemma 5,implies the statement for the concave functional. (cid:3) Let us now characterize improper convex functionals.A continuous convex functional can attain the value + ∞ . Example 9.
Let ϕ : R → R ν be given by ϕ ( x ) = ν if x < , + ∞ if x = 0 , x if x > .ϕ is a continuous convex functional. Proposition 5.
Let ϕ : X → R ν be a convex function on a linear space X . (a) dom − ϕ := { x ∈ dom ϕ | ϕ ( x ) = + ∞} is convex. (b) If ϕ ( x ) = −∞ for some x ∈ X , then ϕ ( λx + (1 − λ ) x ) = −∞ for all x ∈ dom − ϕ, λ ∈ (0 , . (c) If there exists some x ∈ X such that ϕ ( x ) = −∞ , then ϕ ( x ) = −∞ holdsfor all x ∈ icr(dom − ϕ ) . (d) If { x, − x } ⊂ dom ϕ , ϕ ( x ) = −∞ and ϕ (0) = −∞ , then ϕ ( − λx ) = + ∞ forall λ ∈ (0 , . (e) Assume that X is a topological vector space, that ϕ is lower semicontinuouson dom − ϕ and attains the value −∞ . Then ϕ has no finite values.Proof. (a) Consider x , x ∈ dom − ϕ , λ ∈ (0 , x := λx + (1 − λ ) x . There exist t , t ∈ R such that ( x , t ) , ( x , t ) ∈ epi ϕ . Since epi ϕ is convex, we getfor x and t := λt + (1 − λ ) t ∈ R that ( x, t ) ∈ epi ϕ . Hence x ∈ dom ϕ and ϕ ( x ) ≤ t , which implies x ∈ dom − ϕ .(b) Assume x ∈ X with ϕ ( x ) = −∞ and x ∈ dom − ϕ . ⇒ ∀ λ ∈ (0 ,
1) : x λ := λx + (1 − λ ) x ∈ dom − ϕ because of (a). ∃ t ∈ R : ( x , t ) ∈ epi ϕ .Consider an arbitrary λ ∈ (0 , t λ ∈ R . t := λ t λ − − λλ t ∈ R . ⇒ t λ = λt + (1 − λ ) t . ⇒ ( x λ , t λ ) ∈ epi ϕ , because ( x , t ) , ( x , t ) ∈ epi ϕ and epi ϕ is convex. Since ( x λ , t λ ) ∈ epi ϕ for each t λ ∈ R , we get { x λ } × R ⊆ epi ϕ and ϕ ( x λ ) = −∞ for each λ ∈ (0 , x ∈ X with ϕ ( x ) = −∞ and x ∈ icr(dom − ϕ ). ⇒ ∃ ǫ > x := x + ǫ ( x − x ) ∈ dom − ϕ . x = λx + (1 − λ ) x with λ := ǫ ∈ (0 , ϕ ( x ) = −∞ follows from (b).(d) Assume, under the assumptions of (d), that there exists some λ ∈ (0 , ϕ ( − λx ) = + ∞ . ⇒ − λx ∈ dom − ϕ . ⇒ ϕ (0) = −∞ because of (b), acontradiction.(e) Assume x ∈ X with ϕ ( x ) = −∞ and x ∈ dom − ϕ . ⇒ ∀ λ ∈ (0 ,
1) : x λ := λx + (1 − λ ) x ∈ dom − ϕ and ϕ ( x λ ) = −∞ because of (b). ∀ U ∈ N ( x ) ∃ λ ∈ (0 ,
1) : x λ ∈ U . This results in ϕ ( x ) = −∞ because ofthe definition of lower semicontinuity. (cid:3) Let us mention that dom − ϕ is the projection of epi ϕ onto X . Remark 5.
Part (c) of the proposition can be found in [16] , part (c) and (e) for X = R n in [13] , but both authors define convexity with inequality (6.1) and referto the classical framework, where + ∞ is used instead of ν and points with functionvalue + ∞ are excluded from the effective domain. Lemma 6.
Let X be a linear space and ϕ : X → R ν .If ϕ is convex or concave, then each of the following functions has the same property: (a) λϕ with λ ∈ R > , (b) ϕ + c with c ∈ R , (c) g : X → R ν defined by g ( x ) = ϕ ( x − x ) with x ∈ X , (d) g : X → R ν defined by g ( x ) = ϕ ( λx ) with λ ∈ R \ { } . Linearity and Related Algebraic Properties
Usually, a linear function is a function that maps into a vector space. This isnot the case for an extended real-valued function.
Definition 14.
Let X be a linear space and ϕ : X → R ν . ϕ is an affine functional if ϕ is convex and concave.If dom ϕ = X , then ϕ is a linear functional if ϕ is convex and concave and ϕ (0) = 0 holds. Remark 6.
Fenchel [5, p. 59] proved that real-valued affine functionals on finite-dimensional vector spaces defined in the traditional way by an equality are just thefunctionals which are convex and concave.
Obviously, each functional with a nonempty, convex effective domain which isconstant on this domain with the value c is affine and, in the case c = 0, also linear.We get from Definition 14: Lemma 7.
Let X be a linear space and ϕ : X → R . ϕ is an affine functional with ϕ (0) ∈ R if and only if it is the sum of some linearfunctional ϕ l : X → R and some real value. An affine functional is not necessarily the sum of a linear functional and a con-stant value. Example 10.
Define ϕ : R → R by ϕ ( x ) = (cid:26) −∞ if x ≤ , + ∞ if x > . epi ϕ = ( −∞ , × R is convex and closed. hypo ϕ = (0 , + ∞ ) × R is a convex set. ϕ is an affine, lower semicontinuous functional, but not the sum of a linear functionaland some value c ∈ R . We get from Lemma 6:
Lemma 8.
Let X be a linear space and ϕ : X → R ν .If ϕ is affine, then each of the following functions is also affine: (a) λϕ with λ ∈ R , (b) ϕ + c with c ∈ R , (c) g : X → R ν defined by g ( x ) = ϕ ( x − x ) with x ∈ X . (d) g : X → R ν defined by g ( x ) = ϕ ( λx ) with λ ∈ R .If dom ϕ = X and ϕ is linear, then the functions in (a) and (d) are also linear. A functional which is affine and lower semicontinuous on its domain is not nec-essarily continuous.
Example 11.
Define ϕ : R → R ν by ϕ ( x ) = if x = 0 , + ∞ if x ∈ (0 , ,ν if x ∈ R \ [0 , . dom ϕ = [0 , is closed and convex. epi ϕ = { } × [1 , + ∞ ) is convex and closed. hypo ϕ = ( { } × ( −∞ , ∪ ((0 , × R ) is convex, but not closed. ϕ is an improperconvex and concave functional which is lower semicontinuous on its domain, butnot continuous. Let us now prove a statement (cp. [15]) which we will use in the next proposition.
Lemma 9.
Assume that ( X, τ ) is a topological vector space, A ⊂ X , a ∈ A , x ∈ X \ cl A . (a) There exists some t ∈ (0 , such that ta + (1 − t ) x ∈ bd A . (b) There exists some t ∈ (0 , such that ta + (1 − t ) x ∈ bd M A , where bd M A denotes the boundary of A w.r.t. the topology τ M induced by τ on M := { ta + (1 − t ) x | t ∈ [0 , } .Proof. (a) t := inf { t ∈ R + : ta + (1 − t ) x ∈ A } = inf { t ∈ R + : x + t ( a − x ) ∈ A } ≤ a ∈ A .Assume t = 0. ⇒ ∀ U ∈ N ( x ) ∃ t ∈ R + : x + t ( a − x ) ∈ A ∩ U . ⇒ x ∈ cl A ,a contradiction. Thus t > b := t a + (1 − t ) x . The definition of t implies: ∀ V ∈ N ( b ) ∃ t ≥ t ∃ t Let ( X, τ ) be a topological vector space and ϕ : X → R ν . If ϕ is affine and continuous on its effective domain, then it is constant on dom ϕ orproper.Proof. We consider a functional ϕ : X → R ν which is convex, concave and continu-ous on its effective domain, but not proper. Hence part (e) of Proposition 5 impliesthat ϕ does not attain any real value.Assume that ϕ is not constant on dom ϕ . Then there exist x , x ∈ dom ϕ with ϕ ( x ) = −∞ , ϕ ( x ) = + ∞ . M := { tx + (1 − t ) x | t ∈ [0 , } ⊆ dom ϕ since dom ϕ is convex. A := M ∩ dom − ϕ .Since ϕ is continuous and dom − ϕ = { x ∈ dom ϕ | ϕ ( x ) = −∞} , there existssome U ∈ N ( x ) with U ∩ dom − ϕ = ∅ . Thus x / ∈ cl dom − ϕ . Because of x ∈ A and x / ∈ cl A , Lemma 9 implies the existence of some t ∈ (0 , 1] suchthat a := tx + (1 − t ) x ∈ bd M A , where bd M A denotes the boundary of A w.r.t.the topology induced by τ on M . a ∈ dom ϕ since dom ϕ is convex. The continuityof ϕ implies ϕ ( a ) = −∞ by the definition of A and the existence of some neigh-borhood V of a such that ϕ ( x ) = −∞ for all x ∈ V ∩ dom ϕ . Since a ∈ bd M A ,there exists some x ∈ V ∩ M : x / ∈ A . This implies x / ∈ dom − ϕ , a contradiction.Consequently, ϕ is constant on dom ϕ . (cid:3) Corollary 5. Assume that X is a topological vector space and that ϕ : X → R iscontinuous and linear. Then ϕ is finite-valued. A linear functional may be improper without being constant. Example 12. Define ϕ : R → R by ϕ ( x ) = −∞ if x < , if x = 0 , + ∞ if x > . epi ϕ = (( −∞ , × R ) ∪ ( { }× [0 , + ∞ )) and hypo ϕ = ((0 , + ∞ ) × R ) ∪ ( { }× ( −∞ , are convex, but not closed. ϕ is an improper linear functional. Definition 14 is compatible with the usual definition of linear and affine real-valued functions. This will be shown in Theorem 2.Let us first introduce further algebraic properties of extended real-valued func-tions. Definition 15. Let X be a linear space and ϕ : X → R ν . ϕ is said to be (a) positively homogeneous if dom ϕ and epi ϕ are nonempty cones, (b) subadditive if dom ϕ + dom ϕ ⊆ dom ϕ and epi ϕ + epi ϕ ⊆ epi ϕ hold, (c) superadditive if dom ϕ + dom ϕ ⊆ dom ϕ and hypo ϕ + hypo ϕ ⊆ hypo ϕ hold, (d) additive if it is subadditive and superadditive, (e) sublinear if dom ϕ and epi ϕ are nonempty convex cones, (f) odd if dom ϕ = − dom ϕ and ϕ ( − x ) = − ϕ ( x ) is satisfied for all x ∈ dom ϕ , (g) homogeneous if ϕ is positively homogeneous and odd, According to our definition, ϕ (0) ∈ { , −∞} holds for each positively homoge-neous and for each sublinear functional. ϕ is superadditive if and only if − ϕ issubadditive. Lemma 10. Let C be a nonempty subset of a linear space X . C is a convex cone ⇐⇒ λ c + λ c ∈ C for all λ , λ ∈ R + , c , c ∈ C, ⇐⇒ C is a cone and C + C ⊆ C, ⇐⇒ C is convex, C + C ⊆ C and ∈ C. This implies: Lemma 11. Let X be a linear space and ϕ : X → R ν . ϕ is sublinear ⇐⇒ ϕ is convex and positively homogeneous , ⇐⇒ ϕ is subadditive and positively homogeneous , ⇐⇒ ϕ is convex and subadditive and ϕ (0) ≤ . The properties defined in Definition 15 coincide with the usual definitions forreal-valued functions. Proposition 7. Let X be a linear space and ϕ : X → R ν . (a) ϕ is positively homogeneous with ϕ (0) = 0 if and only if dom ϕ is a nonemptycone and ϕ ( λx ) = λϕ ( x ) is satisfied for all λ ∈ R + and x ∈ dom ϕ . (b) A proper function ϕ is subadditive or superadditive if and only if dom ϕ + dom ϕ ⊆ dom ϕ and ϕ ( x + x ) ≤ ϕ ( x ) + ϕ ( x ) or ϕ ( x + x ) ≥ ϕ ( x ) + ϕ ( x ) , respectively , holds for all x , x ∈ dom ϕ . (c) ϕ is homogeneous if and only if dom ϕ is a nonempty cone, dom ϕ = − dom ϕ and ϕ ( λx ) = λϕ ( x ) holds for all λ ∈ R and x ∈ dom ϕ .Proof. (a) (i) Suppose first that ϕ is positively homogeneous, i.e. that dom ϕ andepi ϕ are cones. Assume λ ∈ R + \ { } . Consider first some x ∈ dom ϕ with t := ϕ ( x ) ∈ R . λ · ( x, t ) ∈ epi ϕ since ( x, t ) ∈ epi ϕ . ⇒ ϕ ( λx ) ≤ λt . Suppose ϕ ( λx ) < λt . ⇒ ∃ λ ∈ R with λ < λt : ϕ ( λx ) < λ . ⇒ ( λx, λ ) ∈ epi ϕ .If λ > 0, then λ · ( λx, λ ) ∈ epi ϕ , i.e. ( x, λ λ ) ∈ epi ϕ . ⇒ t = ϕ ( x ) ≤ λ λ ,which implies λt ≤ λ , a contradiction. Thus ϕ ( λx ) = λt = λϕ ( x ).Consider now some x ∈ dom ϕ with ϕ ( x ) = −∞ . ⇒ ( x, t ) ∈ epi ϕ for all t ∈ R . ⇒ ( λx, λt ) ∈ epi ϕ for all t ∈ R since epi ϕ is a cone. ⇒ ϕ ( λx ) = −∞ = λϕ ( x ).Consider finally some x ∈ dom ϕ with ϕ ( x ) = + ∞ . If ϕ ( λx ) = + ∞ , then ϕ ( x ) = ϕ ( λ λx ) = λ ϕ ( λx ) = + ∞ by the above statements, a contradiction.(ii) Assume now that dom ϕ is a cone and ϕ ( λx ) = λϕ ( x ) is satisfied forall λ ∈ R + and x ∈ dom ϕ . If ( x , t ) ∈ epi ϕ , then for each λ ∈ R + : λ x ∈ dom ϕ and ϕ ( λ x ) = λ ϕ ( x ) ≤ λ t , hence ( λ x , λ t ) ∈ epi ϕ .Thus epi ϕ is a cone. (b) (i) Suppose first that ϕ is subadditive, i.e. dom ϕ + dom ϕ ⊆ dom ϕ andepi ϕ +epi ϕ ⊆ epi ϕ . Consider x , x ∈ dom ϕ . ⇒ ( x , ϕ ( x )) , ( x , ϕ ( x )) ∈ epi ϕ . ⇒ ( x , ϕ ( x )) + ( x , ϕ ( x )) ∈ epi ϕ . ⇒ ϕ ( x + x ) ≤ ϕ ( x ) + ϕ ( x ).(ii) Assume that dom ϕ + dom ϕ ⊆ dom ϕ and that ϕ ( x + x ) ≤ ϕ ( x ) + ϕ ( x ) holds for all x , x ∈ dom ϕ . If ( x , t ) , ( x , t ) ∈ epi ϕ , then theassumption yields that x + x ∈ dom ϕ and ϕ ( x + x ) ≤ ϕ ( x ) + ϕ ( x ) ≤ t + t . ⇒ ( x + x , t + t ) ∈ epi ϕ .We have proved part (b) for subadditivity. The statement for a superaddi-tive function ϕ results from the subadditivity of − ϕ .(c) (i) Let us first assume that ϕ is homogeneous, i.e. odd and positivelyhomogeneous. Then dom ϕ is a cone, dom ϕ = − dom ϕ , ϕ ( − x ) = − ϕ ( x )and ϕ ( λx ) = λϕ ( x ) hold for all λ ∈ R + and x ∈ dom ϕ . Consider somearbitrary x ∈ dom ϕ and λ < 0. Then ϕ ( λ x ) = ϕ (( − λ ) · ( − x )) =( − λ ) ϕ ( − x ) = ( − λ ) · ( − ϕ ( x )) = λ ϕ ( x ). Consequently, ϕ ( λx ) = λϕ ( x )holds for all λ ∈ R and x ∈ dom ϕ .(ii) The reverse direction of the equivalence is obvious because of (a). (cid:3) The statement of Proposition 7 (b) cannot be extended to functions which arenot proper. This is illustrated by Example 12. There the function ϕ is additive,but ϕ ( − = ν = ϕ ( − 1) + ϕ (+1). Lemma 12. Let X be a linear space and ϕ : X → R ν . (a) If ϕ is positively homogeneous, additive, odd or homogeneous, then λϕ with λ ∈ R has the same property. (b) If ϕ is subadditive, superadditive or sublinear, then λϕ with λ ∈ R + \ { } has the same property. (c) If ϕ is subadditive, then ϕ + c with c ∈ R , c > , is subadditive. (d) If ϕ is positively homogeneous, subadditive, superadditve, additive, sublin-ear, odd or homogeneous, then g : X → R ν defined by g ( x ) = ϕ ( λx ) with λ ∈ R \ { } has the same property. We are now going to investigate the relationship between additive and linearfunctions. Lemma 13. Let ϕ : X → R ν be a subadditive function on a linear space X . (a) If ϕ (0) = −∞ , then ϕ does not attain any real value. (b) If { x, − x } ⊂ dom ϕ , ϕ ( x ) = −∞ and ϕ (0) = −∞ , then ϕ ( − x ) = + ∞ . (c) If ϕ (0) ∈ R , then ϕ (0) ≥ and ϕ ( nx ) = −∞ for all n ∈ N \ { } and all x ∈ dom ϕ with ϕ ( x ) = −∞ .Proof. (a) Assume ϕ (0) = −∞ and λ := ϕ ( x ) ∈ R for some x ∈ dom ϕ . ⇒ (0 , t ) ∈ epi ϕ for all t ∈ R , ( x, λ ) ∈ epi ϕ . ⇒ (0 , t ) + ( x, λ ) = ( x, t + λ ) ∈ epi ϕ forall t ∈ R . ⇒ ϕ ( x ) = −∞ , a contradiction.(b) Assume that ϕ ( − x ) = + ∞ . ⇒ ∃ λ ∈ R : ( − x, λ ) ∈ epi ϕ . Since ( x, t ) ∈ epi ϕ for all t ∈ R and ϕ is subadditive, we get ( x − x, t + λ ) ∈ epi ϕ for all t ∈ R . Thus ϕ (0) = −∞ , a contradiction.(c) ϕ (0) ∈ R ⇒ (0 , ϕ (0)) ∈ epi ϕ . ⇒ (0 , ϕ (0)) + (0 , ϕ (0)) = (0 , ϕ (0)) ∈ epi ϕ . ⇒ ϕ (0) ≤ ϕ (0). ⇒ ϕ (0) ≥ If ϕ ( x ) = −∞ , then ( x, t ) ∈ epi ϕ for all t ∈ R . ⇒ ( nx, nt ) ∈ epi ϕ for all t ∈ R , n ∈ N \ { } . ⇒ ϕ ( nx ) = −∞ for all n ∈ N \ { } . (cid:3) Proposition 8. Let ϕ : X → R ν be an additive function on a linear space X with ∈ dom ϕ . (a) If ϕ (0) = 0 , then ϕ does not attain any real value. (b) If ϕ (0) = 0 , then ϕ is odd on dom ϕ ∩ ( − dom ϕ ) and ϕ ( tx ) = tϕ ( x ) for all t ∈ Q , x ∈ dom ϕ ∩ ( − dom ϕ ) .Proof. From Lemma 13 we get:If ϕ (0) / ∈ R , then ϕ does not attain any real value.If ϕ (0) ∈ R , then ϕ (0) = 0.This implies (a).Assume now ϕ (0) = 0. Lemma 13 implies for x ∈ dom ϕ ∩ ( − dom ϕ ): ϕ ( x ) = −∞ ifand only if ϕ ( − x ) = + ∞ . Consider now some arbitrary x ∈ dom ϕ ∩ ( − dom ϕ ) with ϕ ( x ) ∈ R . ⇒ ( x, ϕ ( x )), ( − x, ϕ ( − x )) ∈ epi ϕ ∩ hypo ϕ . ⇒ ( x, ϕ ( x )) + ( − x, ϕ ( − x )) =(0 , ϕ ( x )+ ϕ ( − x )) ∈ epi ϕ ∩ hypo ϕ . ⇒ ϕ (0) = ϕ ( x )+ ϕ ( − x ). ⇒ ϕ ( − x ) = − ϕ ( x ).Thus ϕ is odd on dom ϕ ∩ ( − dom ϕ ).Lemma 13 implies ϕ ( nx ) = nϕ ( x ) for all n ∈ N and all x ∈ dom ϕ with ϕ ( x ) / ∈ R .Consider now some x ∈ dom ϕ with ϕ ( x ) ∈ R . ⇒ ( x, ϕ ( x )) ∈ epi ϕ ∩ hypo ϕ . ⇒ ( nx, nϕ ( x )) ∈ epi ϕ ∩ hypo ϕ . ⇒ ϕ ( nx ) = nϕ ( x ) for all n ∈ N . Thus ϕ ( nx ) = nϕ ( x ) for all n ∈ N , x ∈ dom ϕ . For t ∈ − N , x ∈ dom ϕ ∩ ( − dom ϕ ) we get ϕ ( tx ) = ϕ (( − t )( − x )) = − tϕ ( − x ) = − t · ( − ϕ ( x )) = tϕ ( x ). Consider q ∈ Z \ { } , x ∈ dom ϕ ∩ ( − dom ϕ ). Then ϕ ( x ) = ϕ ( q q x ) = qϕ ( q x ). ⇒ ϕ ( q x ) = q ϕ ( x ).Consequently, ϕ ( pq x ) = pq ϕ ( x ) for all x ∈ dom ϕ, p ∈ N , q ∈ Z \ { } . (cid:3) Theorem 2. Let X be a linear space and ϕ : X → R . (1) ϕ is linear if and only if ϕ is homogeneous and additive. (2) ϕ is linear if and only if ϕ and − ϕ are sublinear.Proof. (a) We suppose that ϕ is linear.Assume that ϕ is not odd. ⇒ ∃ x ∈ X : ϕ ( − x ) = − ϕ ( x ). Proposition5(d) implies ϕ ( x ) = −∞ and ϕ ( − x ) = −∞ . Applying Proposition 5(d)to the convex function − ϕ , we get ϕ ( x ) = + ∞ and ϕ ( − x ) = + ∞ . Thus t := ϕ ( x ) ∈ R , t := ϕ ( − x ) ∈ R . ⇒ ( x, t ) , ( − x, t ) ∈ epi ϕ ∩ hypo ϕ . ⇒ ( ( x − x ) , ( t + t )) ∈ epi ϕ ∩ hypo ϕ . ⇒ ϕ (0) = ( t + t ). ⇒ t = − t , a contradiction. Hence ϕ is odd.If ϕ ( x ) = −∞ , then Proposition 5(b) implies ϕ ( λx ) = −∞ for all λ ∈ (0 , ϕ ( x ) = + ∞ and we apply Proposition 5(b) to − ϕ , we get ϕ ( λx ) = + ∞ for all λ ∈ (0 , x ∈ X with t := ϕ ( x ) ∈ R . ⇒ (0 , , ( x, t ) ∈ epi ϕ ∩ hypo ϕ . ⇒ ((1 − λ ) · λx, (1 − λ ) · λt ) ∈ epi ϕ ∩ hypo ϕ for all λ ∈ [0 , ⇒ ( λx, λt ) ∈ epi ϕ ∩ hypo ϕ for all λ ∈ [0 , ⇒ ϕ ( λx ) = λt for all λ ∈ [0 , ϕ ( λx ) = λϕ ( x ) for all x ∈ X , λ ∈ [0 , x ∈ X and λ > ⇒ λ ∈ (0 , ⇒ ϕ ( x ) = ϕ ( λ ( λx )) = λ ϕ ( λx ). ⇒ ϕ ( λx ) = λϕ ( x ). Hence ϕ ( λx ) = λϕ ( x ) for all λ ∈ R + . Since ϕ is odd, we get ϕ ( λx ) = λϕ ( x ) for all λ ∈ R . Thus ϕ is homogeneous byProposition 7.(b) (a) implies part (2) of our assertion because of Lemma 11. (c) Applying Lemma 11 to ϕ and − ϕ implies the additivity of each linearfunction ϕ .(d) If ϕ is homogeneous and additive, then ϕ is linear by (2) and Lemma 11. (cid:3) Thus for real-valued functions, our definition of linearity coincides with the usualone. Corollary 6. Let X be a linear space and ϕ : X → R .Then ϕ is linear if and only if ϕ ( λ x + λ x ) = λ ϕ ( x ) + λ ϕ ( x ) holds for all x , x ∈ X and λ , λ ∈ R . Theorem 3. Let X be a topological vector space and ϕ : X → R be continuous.Then ϕ is linear if and only if ϕ is additive and ϕ (0) = 0 .Proof. Assume that ϕ is additive and ϕ (0) = 0, but that ϕ is not linear. Then ϕ is not homogeneous by Theorem 2. ⇒ ∃ x ∈ X ∃ λ ∈ R : ϕ ( λx ) = λϕ ( x ). ⇒ There exist neighborhoods V of ϕ ( λx ) and V of λϕ ( x ) such that V ∩ V = ∅ .Since ϕ is continuous, the function ϕ ( λx ) is a continuous function of x . Thusthere exists some neighborhood U of x such that for each x ∈ U : ϕ ( λx ) ∈ V .There exists some p ∈ (0 , 1) such that for all t ∈ ( p, tx ∈ U . Considersome sequence ( q n ) n ∈ N of rational numbers with q n < λ for each n ∈ N whichconverges to λ . ⇒ ∃ n ∈ N ∀ n > n : q n λ ∈ ( p, ⇒ ∀ n > n : q n λ x ∈ U . ⇒ ∀ n > n : q n ϕ ( x ) = ϕ ( q n x ) = ϕ ( λ q n λ x ) ∈ V . ⇒ ∀ n > n : q n ϕ ( x ) V . ⇒ q n ϕ ( x ) does not converge to λϕ ( x ) for n → ∞ , a contradiction. (cid:3) Usage of Results from Convex Analysis Many essential facts for extended real-valued functions have been proved in con-vex analysis and can be applied to the extended real-valued functions in our ap-proach by making use of the following remarks.There is a one-to-one correspondence between extended real-valued functions inconvex analysis and those extended real-valued functions in the unified approachwhich do not attain the value + ∞ . This will be shown in Section 8.2. In Section8.3, the way of transferring results from convex analysis to functions in the unifiedapproach which can also attain the value + ∞ is pointed out. Since we have notdefined all properties which are of interest for functionals, we start in Section 8.1with an explanation of how the definition of notions and properties for functionsworks in the unified approach.8.1. Definition of notions and properties for functions in the unified ap-proach. By defining notions and properties in the unified approach, one simply has to takeinto consideration that the value ν stands for ”not being defined” or ”not beingfeasible”.A function in the unified approach has a property if and only if it has it on itsdomain, where the domain has to fulfill those conditions which are essential for thedefined property. The definition for the property on the domain can be taken fromconvex analysis. This is illustrated by the previous sections and by the followingdefinitions where the second one is, e.g., of importance in vector optimization. Definition 16. Let X be a real normed space, ϕ : X → R ν and X ⊆ dom ϕ with X = ∅ . ϕ is called Lipschitz continuous on X if ϕ is finite-valued on X and if there existssome L ∈ R + such that | ϕ ( x ) − ϕ ( x ) | ≤ L k x − x k for all x , x ∈ X . ϕ is said to be locally Lipschitz continuous on X if for each x ∈ X there existssome neighborhood U of x such that ϕ is Lipschitz continuous on U ∩ X . ϕ is locallyLipschitz continuous or Lipschitz continuous if ϕ is a proper functional which islocally Lipschitz continuous or Lipschitz continuous, respectively, on dom ϕ . Definition 17. Let X be a linear space, B ⊆ X and ϕ : X → R ν , X ⊆ dom ϕ . ϕ is said to be (a) B -monotone on X if x , x ∈ X and x − x ∈ B imply ϕ ( x ) ≤ ϕ ( x ) , (b) strictly B -monotone on X if x , x ∈ X and x − x ∈ B \ { } imply ϕ ( x ) < ϕ ( x ) . ϕ is said to be B -monotone or strictly B -monotone if it is B -monotone or strictly B -monotone, respectively, on dom ϕ . In variational analysis, basic definitions can be extended to the unified approachas follows. Definition 18. Suppose that X is a separated locally convex space with topologicaldual space X ∗ and f : X → R ν . The (Fenchel) subdifferential ∂f ( x ) of f at x ∈ X is the set { x ∗ ∈ X ∗ | ∀ x ∈ dom f : x ∗ ( x − x ) ≤ f ( x ) − f ( x ) } if f ( x ) ∈ R , and the empty set if f ( x ) / ∈ R .The (Fenchel) conjugate of f is the function f ∗ : X ∗ → R ν defined by f ∗ ( x ∗ ) := sup { x ∗ ( x ) − f ( x ) | x ∈ dom f } . The support function σ A : X ∗ → R ν of a set A ⊆ X is defined by σ A ( x ∗ ) := sup { x ∗ ( a ) | a ∈ A } . The set { x ∗ ∈ X ∗ | σ A ( x ∗ ) ∈ R } is called the barrier cone bar A of A . Usage of results from convex analysis for functions which do notattain the value + ∞ in the unified approach. If a functional attains only values from R ∪ {−∞} on the entire space, then thenotions for and properties of this function are the same in convex analysis and inour unified approach.If a statement in convex analysis is given for some functional f cxa : X → R , wecan transfer this statement to f ua : X → R ν in the unified approach given by f ua ( x ) = (cid:26) f cxa ( x ) if f cxa ( x ) = + ∞ ,ν if f cxa ( x ) = + ∞ , using the following interdependencies.Note that f ua does not attain the value + ∞ .I. Notions and properties which are similar for f cxa and f ua (a) The (effective) domain of f cxa in the terminology of convex analysisis, in the unified approach, the setdom f ua = { x ∈ X | f ua ( x ) ∈ R ∪ {−∞}} = { x ∈ X | f cxa ( x ) ∈ R ∪ {−∞}} = dom − f cxa . (b) f ua is proper in the terminology of the unified approach ⇔ f cxa is proper in the terminology of convex analysis ⇔ f ua attains some real value, but not the value −∞ on X ⇔ f cxa attains some real value, but not the value −∞ on X .(c) f ua is finite-valued in the terminology of the unified approach ⇔ f cxa is finite-valued in the terminology of convex analysis ⇔ f ua attains only real values on X ⇔ f cxa attains only real values on X .(d) epi f ua = epi f cxa .Our definition of the epigraph applied to f cxa coincides with that inconvex analysis.(e) For each x ∈ dom − f cxa , f cxa is lower semicontinuous at x if and onlyif f ua is lower semicontinuous at x .The notion of lower semicontinuity of f cxa at x ∈ dom − f cxa coincidesin convex analysis and in the unified approach.(f) f cxa is convex if and only if f ua is convex.Our definition of convexity applied to f cxa coincides with that in con-vex analysis.(g) For f cxa , the definitions of B -monotonicity, strict B -monotonicity, Lip-schitz continuity and local Lipschitz continuity on a nonempty subsetof dom − f cxa in the unified approach are the same as in convex ana-lysis. f cxa has one of these properties if and only if f ua has it.(h) ∂f ua ( x ) = ∂f cxa ( x ) for each x ∈ X if X is a separated locally convexspace.Our definition of the subdifferential applied to f cxa coincides with thatin convex analysis.II. Notions and properties which are different for f cxa and f ua (j) For each nonempty set X ⊆ dom f ua , the notions infimum, supremum,maximum, minimum, upper bound and lower bound on X have thesame contents for f cxa and f ua . In this case, the definitions of thesenotions coincide for f cxa with the usual definitions in convex analysis.If X = ∅ , we get in convex analysis, inf x ∈ X f cxa ( x ) = + ∞ andsup x ∈ X f cxa ( x ) = −∞ ,and in the unified approach, inf x ∈ X f ua ( x ) = sup x ∈ X f ua ( x ) = ν .If X \ dom − f cxa = ∅ , we get in convex analysis,inf x ∈ X f cxa ( x ) = inf x ∈ dom f cxa f cxa ( x ) and sup x ∈ X f cxa ( x ) = + ∞ ,and in the unified approach, inf x ∈ X f ua ( x ) = inf x ∈ dom f ua f ua ( x )and sup x ∈ X f ua ( x ) = sup x ∈ dom f ua f ua ( x ).(k) f cxa is continuous at x ∈ dom − f cxa if and only if x ∈ int dom f ua and f ua is continuous at x .The analogous statement for upper semicontinuity instead of continu-ity holds as well. At each x ∈ dom − f cxa \ int(dom − f cxa ), f cxa is not upper semicontin-uous and hence not continuous according to the definition in convexanalysis as well as to the definition in the unified approach, but f ua may be upper semicontinuous or continuous at x in the unified ap-proach.At x ∈ int(dom − f cxa ), the notions of continuity and upper semiconti-nuity of f cxa in convex analysis are the same as in the unified approach.Let us point out that these differences are advantages of the unified ap-proach.8.3. Usage of results from convex analysis for functions which attain thevalue + ∞ in the unified approach. Statements from convex analysis for functionals f cxa : X → R can be transferred tostatements for functions ϕ : X → R ν in the unified approach which can also attainthe value + ∞ according to the following rules.Replace by X dom ϕ dom f cxa dom − ϕ finite-valued properproper ϕ attains some real value and not the value −∞ The sets on which f cxa has a certain property have to be adapted to ϕ accordingto this table. If the set is X , those conditions which the whole space X automaticallyfulfills have to be taken into consideration.9. Final Remarks Our theory serves as a basis for extending functions with values in R to theentire space and handling them in a way which is not unilateral. It offers thesame possibilities as the approach in convex analysis, but avoids several of itsdisadvantages and delivers a calculus for functions not depending on the purposethey are used for.All results of the unified approach can immediately be applied in the classicalframework of convex analysis since each function ϕ : X → R also maps into R ν .Replace (unified approach) by (classical approach)dom ϕ X dom − ϕ dom ϕ proper finite-valuedLet us finally mention that the application of ν is not restricted to R . ν can beconsidered in combination with each space Y . The concept presented in this papercan easily be adapted to Y ν := Y ∪ { ν } and to functions with values in Y ν . Supposesome function f : C → Y with C being a subset of some space X and Y beinga linear space. f could be extended to X by the function value ν . An indicatorfunction ι C : X → Y ν can be defined in the same way as above, where 0 stands forthe element of Y which is neutral w.r.t. addition. 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