Semicontinuities of Multifunctions and Functions

Transcription

1 Chapter 4 Semicontinuities of Multifunctions and Functions The notion of the continuity of functions is certainly well known to the reader. This topological notion plays an important role also for multifunctions. It is not surprising that for general objects as multifunctions are, several types of continuity can be introduced and these continuities often but not always coincide when applied to a function. 4.1 Lower and upper semicontinuity We start the study of various types of continuity of multifunctions by reminding the well known continuity conditions of functions between topological spaces (X, τ) and (Y, σ). A function f : (X, τ) (Y, σ) is continuous if any of the following mutually equivalent conditions is satisfied (in these conditions U is an arbitrary subset of X and V is an arbitrary subset of Y) (a1) V is open in (Y, σ) f 1 [V] is open in (X, τ) (a2) f [U] f [U] (a3) f 1 [ V] f 1 [V] (a4) x X: V is a neighbourhood of f (x) f 1 [V] is a neighbourhood of x (b1) V is closed in (Y, σ) f 1 [V] is closed in (X, τ) (b2) f 1 [V] f 1 [V] (b3) f + [ V] f + [V] 54

2 4.1 Lower and upper semicontinuity 55 (b4) x X: V is a neighbourhood of f (x) f + [V] is a neighbourhood of x These conditions need not be equivalent for a multifunction, however, the conditions (a1)-(a4) stay mutually equivalent and also conditions (b1)-(b4) are mutually equivalent for an arbitrary multifunction. This leads us to two definitions of semi-continuity of multifunctions. As we will need also a local concept of semi-continuity of multifunction, we introduce it by the following definition. Definition 4.1. (a) We say that a multifunction F : X Y is lower [upper] semi-continuous at a point x X if for every open set V Y exists an open set U X such that F(x) V x U F 1 [V] [ F(x) V x U F + [V] ]. (b) We say that F is a lower [upper] semi-continuous multifunction on a set M X if it is lower [upper] semi-continuous at every point from M. If F is lower [upper] semi-continuous at every point of its domain X, then we simply say that F is a lower [upper] semi-continuous multifunction and we will use the abbreviation l.s.c [u.s.c]. (c) We say that F is continuous multifunction (at a point x X) if it is both upper and lower semi-continuous (at the point x). Using the neighbourhood concept of a set (of a point), the semi-continuity can be expressed in the following way: F is upper semi-continuous at a point x if the upper pre-image F + [V] of any neighbourhood V of the full image F(x) is a neighbourhood of the point x. Similarly F is lower semi-continuous at a point x if the lower pre-image F 1 [V] of any neighbourhood V of each point y F(x) is a neighbourhood of the point x. Proposition 4.2. For any multifunction F : any sets U X and V Y: X Y the following statements are equivalent for (i) F is l.s.c. at a point x 0 (ii) F(x 0 ) V F 1 [ V] is a neighbourhood of x 0 (iii) x 0 F 1 [ V] x 0 F 1 [V] (iv) x 0 U F(x 0 ) F[U] Proof. (i) (ii) because V is an open neighbourhood of a chosen point y F(x 0 ) V and F 1 [ V] is a neighbourhood of x 0 iff there exists an open set O X such that x 0 O F 1 [ V]. (ii) (i) is straightforward since for an open set V Y we have V = V and F 1 [V] is a 55

3 4.1 Lower and upper semicontinuity 56 neighbourhood of x 0 iff x 0 F 1 [V]. (ii) (iii) since F 1 [ V] is a neighbourhood of x 0 iff x 0 F 1 [ V] F 1 [ V] F 1 [V]. F 1 [V] and obviously (iii) (ii) because if the implication in (iii) holds for any set V Y then it holds true also for any open set V = V, i.e. x 0 F 1 [ V] x 0 F 1 [ V], but V = V, and x 0 F 1 [ V] is equivalent to say that F 1 [ V] is a neighbourhood of x 0. (iii) (iv) for each y F(x 0 ) and any neighbourhood V of y there exists a neighbourhood O of x 0 such that for any x the implication holds true x O F(x) V. Since x U, we have O U, and so there exists x O U with F(x ) V and F(x ) F[U]. This gives V F[U] and hence y F[U]. (iv) (iii) Let us proceed by contradiction, supposing the implication (iii) were not true. Then there exists y F(x 0 ) and its neighbourhood V y such that in each neighbourhood N of x 0 there exists a point x N N such that F(x N ) V =. Hence the set M N := {x N F(x) V = }, and if we consider the set U := {M N N is a neighbourhood of x} then x 0 U. But y F[U] because x U : F(x) V =, and consequently = {F(x) V x U} = V {F(x) x U} = V F[U]. It gives a contradiction with the conclusion in (iv), namely F(x 0 ) F[U]. THEOREM 4.3. For every multifunction F : X Y the following statements are equivalent (a1) F is l.s.c., (a2) V Y F 1 [ V] is open, 56

4 4.1 Lower and upper semicontinuity 57 (a3) V Y F 1 [ V] F 1 [V], (a4) U X F[U] F[U], (a5) The associated set-valued function ˆF : X P(Y) is continuous when we consider the lower topology L on P(Y), generated by the subbase {P(Y) \ P(K) K closed in Y}. Proof. Immediate consequence of Proposition 4.2 and the fact that F is l.s.c. iff it is l.s.c. at each point x 0 X. Proposition 4.4. For every multifunction F : X Y the following statements are equivalent for any set V Y: (i) F is u.s.c. at a point x 0, (ii) F(x 0 ) V F + [ V] is a neighbourhood of x 0, (iii) x 0 F + [ V] x 0 F + [V], (iv) x 0 F 1 [V] x 0 F 1 [V]. Proof. (i) (ii) (iii) is easy to prove from the definition of upper semicontinuity and it is similar to the proof of similar equivalences in Proposition 4.2. (iii) (iv) let us suppose there exists V Y verifying Then F(x 0 ) V = which means that x 0 F 1 [V] \ F 1 [V]. F(x 0 ) Y \ V Y \ V. Using (iii) there exists a neighbourhood U of x 0 with the property: U F + [Y \ V] = X \ F 1 [V], i.e. U F 1 [V] = which contradicts the assumption x 0 F 1 [V]. (iv) (iii) let us choose arbitrary set V Y such that x 0 F + [ V]. Then and hence x 0 F 1 [Y \ V]. Using (iv) we have F(x 0 ) (Y \ V) =, x 0 F 1 [Y \ V], 57

5 4.1 Lower and upper semicontinuity 58 and therefore there exists a neighbourhood U of x 0 satisfying U F 1 [Y \ V] =. Then the following relationships hold x 0 U X \ F 1 [Y \ V] = F + [ V] F + [V], and this implies x 0 F + [V]. THEOREM 4.5. For any multifunction F : X Y the following statements are equivalent (b1) F is u.s.c., (b2) V Y F + [ V] is open in X, (b3) V Y F + [ V] F + [V] (b4) V Y F 1 [V] F 1 [V], consequently F 1 [V] is closed in X, (b5) The associated set-valued function ˆF : X P(Y) is continuous if we consider the upper topology U on P(Y), generated by the base {P(G) G is open in Y}. Proof. An immediate consequence of Proposition 4.4 and the fact that u.s.c. means u.s.c. at each point x 0 X. Remark 4.6. Using our convention that the notation of a multifunction in the form F : X Y means that D F = X and hence F is in fact a correspondence of X into Y, the above definition of semicontinuity was done for correspondences. It can be applied without change to any multifunction F from X to Y, i.e. F X Y with F(x) = when x X \ D F, i.e. to the associated set-valued function ˆF : X P(Y). In such case obviously F is l.s.c. at any point x 0 X \ D F and is u.s.c. at x 0 X \ D F only if x 0 is an interior point of X \ D F, F cannot be u.s.c. at any point x 0 D F \ D F which is a point in the boundary of D F not in D F. So if D F is closed in X then F is continuous (i.e. lower as well as upper semicontinuous) at any point x 0 X \ D F. Because of this it is more common to define the notion of semicontinuity for a general multifunction F X Y in the way where the topological notions are relativised to the subspace D F of X (when applicable). The following examples show that the notions of lower and upper semicontinuity are independent and also give good graphic intuition for these notions. Example 4.7. (a) The correspondence F 1 : R R defined by [ 1, 1], if x 0 F 1 (x) := {0} if x = 0 58

6 4.1 Lower and upper semicontinuity 59 is lower semicontinuous at x = 0 but it is not upper semicontinuous at that point. In the Cartesian graph of F 1 shown in the left panel of Figure 4.1 we can see that big values F 1 (x) = [ 1, 1] can shrink down rapidly to a small value F 1 (0) = {0} and the lower semicontinuity will be preserved. (b) The correspondence F 2 : R R defined by {0} if x 0 F 2 (x) := [ 1, 1] if x = 0 is upper semicontinuous at x = 0 but it is not lower semicontinuous at that point. In the Cartesian graph of F 2 shown in the right panel of 4.1 we can see that small values F 2 (x) = {0} can rapidly grow up to a big value F 2 (0) = [ 1, 1] and the upper semicontinuity will be preserved Figure 4.1: Left (right) panel shows the Cartesian graph of the multifunction F 1 (F 2 ) from Example 4.7. The above mentioned results may raise an impression in the reader that there are seemingly parallel statements for lower semicontinuous and upper semicontinuous multifunctions. But these continuity concepts are far from being equally flexible constructs. They are not. For non-compact valued multifunctions, upper semicontinuity is difficult to achieve. G. Choquet showed in his paper [8] that the vertical line multifunction F : R R 2, F(x) := {x} R which intuitively ought to satisfy any reasonable continuity requirement fails to be u.s.c. at any point. The characterization of semicontinuity by net convergence is another example of asymmetry between lower and upper semicontinuity. Continuity at a point of a function from topological space X to topological space Y can be described by net convergence in the form that for any net (x ι ) ι D x 0 we have f (x ι ) f (x 0 ). If we want to extend this characterization to set-valued maps we must introduce the notion of convergence for nets of sets. Definition 4.8. (a) Reflexive and transitive relation on a nonempty set D is called a preordering and the pair (D, ) is called a preordered set. (b) If is a preorder on D and A, E D we define the set of upper [lower] bounds of the set A in 59

7 4.1 Lower and upper semicontinuity 60 the set E by A (E) := {d E a A : d a} [ A (E) := {d E a A : a d} ]. When E = D we use simpler notation A and A respectively, and also for d D, {d} =: d and {d} =: d. (c) The preordered set (D, ) is called a directed set if any two element set {d 1, d 2 } D has an upper bound in D, i.e. {d 1, d 2 }. The subset B D is called a residual set in (D, ) if C D is called a cofinal set in D if d 0 D : d 0 B. d D : d C, i.e. if each element from D has an upper bound in C, or d (C). Example 4.9. (a) Both sets R and N are directed sets with the natural ordering relation. Moreover N is cofinal in R (but not vice-versa). (b) In any topological space (X, τ) the open neighbourhood base at a given point x is a directed set by the inclusion, i.e. τ(x) := {U τ x U} U 1, U 2 τ(x) U τ(x) U U 1 U 2. Some useful properties of residual and cofinal sets are given in Exercise 4.1. Definition Let (D, ) be a directed set. Then any mapping s : D Y, is called a net in Y. The value s(d) is denoted s d and the net s itself is denoted (s d : d D) or just (s d ) when D is understood by the context. When s is a set-valued mapping i.e. Y P(X), it is called a net of subsets of X. From Example 4.9(a) we can conclude that any sequence s : N X can be considered as a net when N is directed by the natural order. There are plenty of other preorders which can direct the set N (for example, or the divisibility relation a b k N : b = ka) but when nothing is specified about order in N we will always suppose the natural order on N. An important notion is a subnet of the given net just like a subsequence is an important notion for the sequences. We know that a sequence b = (b n ) n N is a subsequence of the sequence a = (a n ) n N iff there exists an increasing sequence k : N N such that b n = a kn i.e. b = a k. For nets the situation is a little bit complicated since domains of different nets can be different directed sets. But the scheme from subsequences can be maintained if we replace the increasing sequence k by a more general notion of a residual mapping. 60

8 4.1 Lower and upper semicontinuity 61 Definition Let (D, D ) and (E, E ) be directed sets. Then the mapping r : E D is called a residual mapping if the pre-image of any residual subset in D is a residual subset in E. The following proposition gives several characterizations of residual mappings. Proposition Each of the following statements is equivalent to the fact that the mapping r : (E, E ) (D, D ) is residual: (i) d D : r 1 [d ] is residual in E, (ii) d D e E : e r 1 [d ], (iii) d D e E e E e : r(e ) D d, (iv) C E cofinal: r[c] is cofinal in D. Because of this property, some authors call r a cofinal mapping. Proof. The reader is asked to do it in Exercise 4.2. Using the notion of residual mapping we can introduce the notion of a subnet of the given net. Definition A net t : (E, E ) X is called a subnet of the net s : (D, D ) X, and we write t s, if there exists a residual map r : E D such that t = s r i.e. the diagram shown in Figure 4.2 commutes. D s X r E t Figure 4.2: Now we can define two types of limits for a net of subsets of the topological space (X, τ). Definition Let (A ι ) ι I be a net of subsets of the topological space (X, τ). A point x 0 X is called (a) a limit point of the net (A ι ) if each neighbourhood of x 0 intersects (A ι ) for all ι in some residual subset of I. We denote the set of all limit points of (A ι ) by Li(A ι : ι I) or just LiA ι and call it a residual limit of (A ι ) (some authors call it a lower topological limit or a lower closed limit). Hence we can write x 0 Li ι I A ι U τ(x 0 ) ι 0 : ι ι 0 : U A ι. 61

9 4.1 Lower and upper semicontinuity 62 (b) A point x 1 X is called a cluster point of the net (A ι ) if each neighbourhood of x 1 intersects (A ι ) for all ι in some cofinal subset of I. We denote by Ls(A ι : ι I) or just by LsA ι the set of all cluster points of (A ι ) and call it a cofinal limit of (A ι ) (alternative terminology is an upper topological limit or upper closed limit). (c) We say that a net (A ι X) ι I is Kuratowski-Painlevé convergent to a set A X, and we write A = K lim A ι, provided LiA ι = LsA ι = A. Since obviously LiA ι LsA ι to verify that A = K lim A ι, it is necessary and sufficient to prove the inclusions A LiA ι, LsA ι A. Before giving a net characterization of the lower semicontinuity of multifunctions, we prove the following characterization of the lower and upper limits. Proposition For any net of sets (A ι ) ι I in a topological space (X, τ) the following equivalences are true (a) x 0 LiA ι ι 0 I ι ι 0 x ι A ι : (x ι ) ι ι0 x 0 (b) x 0 LsA ι ι I ι ι x ι A ι : (x ι ) x 0 Proof. We prove (a) and leave to the reader the similar proof of (b). ( ) If U N(x 0 ) then ι U I ι ι U : x ι U. Then ι 1 I : ι 1 ι 0, ι U and for any ι ι 1 we have x ι U A ι, hence x 0 LiA ι. ( ) This implication is true if LiA ι =. So suppose that LiA ι and then there exists a neighbourhood V = X of x 0 and consequently there exists ι 0 I such that ι ι 0 : A ι X = A ι. Then the set D := {(ι, V) I N(x 0 ) A ι V } is directed by the relation (ι, V) (ι, V ) ι ι & V V, and we can define the net t := (x ι,v A ι V : (ι, V) D) which converges to x 0 because for any neighbourhood W x 0 ι W I ι ι W : A ι W. Hence (ι W, W) D and for each (ι, V) (ι W, W) we have x ι,v A ι V A ι W W. 62

10 4.1 Lower and upper semicontinuity 63 Now we are going to construct a subnet s = (x ι A ι ) ι ι0 of the net t. Consider the first projection which maps D on the residual subset of I. π : D I, (ι, V) ι, π[d] = {ι I V N(x 0 ) : A ι V } ι 0. Consider any selection σ of the inverse correspondence π 1 i.e. σ : π[d] D, σ π 1. σ is a residual map because for any residual set R D we have σ 1 [R] σ 1 [(ι R, V R ) ] = ι R, so σ 1 [R] is residual in π[d] and hence σ is residual map. Now it suffices to put s = t σ to obtain a subset of t which consequently converges to x 0 and has the property that ι ι 0 : s(ι) A ι. Remark We can formulate the characterization of lower and upper closed limits in the more convenient way: x 0 LiA ι if there exists a residual selection net (x ι A ι ) ι ι0 x 0, x 0 LsA ι iff there exists a cofinal selection net (x ι A ι ) ι K x 0. THEOREM The multifunction F : X Y is lower semicontinuous at a point x 0 X iff for each net (x ι X) ι I convergent to x 0 we have F(x 0 ) LiF(x ι ). Proof. ( ) Let y F(x 0 ) and V be a neighbourhood of y. Then F 1 [V] is a neighbourhood of x 0 and so ι 0 I ι ι 0 : x ι F 1 [V], which gives V F(x ι ) ( ι ι 0 ), but it means that y LiF(x ι ). ( ) If F were not lower semicontinuous at x 0 then there exist y F(x 0 ) and a neighbourhood V of y such that F 1 [V] is not a neighbourhood of x 0. Therefore for each neighbourhood U N(x 0 ) we have U F 1 [V] i.e. U (X \ F 1 [V]), 63

11 4.1 Lower and upper semicontinuity 64 and hence we can choose x U U with F(x U ) Y \ V. If we consider the directed set (N(x 0 ), ) we have constructed the net (x U : U N(x 0 )) x 0, with y LiF(x U ). Hence F(x 0 ) LiF(x U ) which contradicts the assumption. Corollary The multifunction F : X Y is lower semicontinuous at x 0 X iff for each net (x ι ) ι I x 0 and each y F(x 0 ) there exists ι 0 I and a net (y ι F(x ι )) ι ι0 y. Proof. It is a direct consequence of the previous theorem and the following characterization of limit points of any net of sets (A ι ) ι I. In the case of metrizable spaces X, Y we obtain from Corollary 4.18 the following characterization of the lower semicontinuity which in many texts is taken as a definition of the lower semicontinuity. THEOREM Let X, Y be metrizable spaces. A multifunction F : X Y is lower semicontinuous at a point x 0 X iff for any sequence (x n ) n 1 x 0 and for any y F(x 0 ) there exist n 0 N and a sequence (y n F(x n )) n n0 converging to y. If F is a correspondence, i.e. x X : F(x), then we can put n 0 = 1. Now we proceed to the characterization of the upper semicontinuity by using nets convergence. From Theorem 4.15 one could think that upper semicontinuity of a multifunction F at a point x 0 could be characterized by the dual notion to the lower closed limit namely by the upper closed limit: if x i x 0 then F(x 0 ) Ls i N F(x i ). But this intuitive impression is false! Such assertion gives a new type of semicontinuity which is called outer semicontinuity which, in general, is different from both lower and upper semicontinuity and especially is equivalent with closedness of the graph of the multifunction. We will study it in the next section. Next theorem gives a net characterization of the local upper semicontinuity. THEOREM A multifunction F : X Y is upper semicontinuous at a point x 0 X iff for any net (x ι : ι I) in X which converges to x 0, and every open set V in Y with F(x 0 ) V, F(x i ) V for sufficiently large ι. It is equivalent with the convergence F(x ι ) F(x 0 ) in the upper topology τ u on P(Y). Proof. ( ) To prove the "only if" part of the theorem suppose (x ι : ι I) x 0. Then for every open V F(x 0 ), F + [V] is a neighbourhood of x 0 and s ι F + [V] for sufficiently large ι ι 0. Consequently, F(x ι ) V for all ι ι 0. 64

12 4.2 Outer semicontinuity 65 ( ) To prove the "if" part of the theorem let us suppose F were not u.s.c. at x 0. Then there exists an open set V F(x 0 ) such that F + [V] is not a neighbourhood of x 0. It can happen only if for each neighbourhood U N(x 0 ) there is x U U such that F(x U ) (Y \ V). ( ) When considering the neighbourhood family N(x 0 ) directed by the inclusion we have constructed a net (x U : U N(x 0 )) converging to x 0 and using the assumption of the "if" part we get U 0 N(x 0 ) U U 0 : F(x U ) V, but this is a contradiction with ( ). 4.2 Outer semicontinuity We have seen that upper semicontinuity is not quite a dual notion to lower semicontinuity, if using characterization of these notions by nets convergence. Outer semicontinuity is such a notion whose global version amounts to a closed graph when the target space is Hausdorff. We start with the local definition described in terms of an upper closed limit. Definition A multifunction F : X Y is outer semicontinuous (o.s.c) at a point x 0 X provided whenever (x ι : ι I) is a net in X convergent to x 0 then the net (F(x ι ) : ι I) of subsets of Y cofinaly converges to F(x 0 ), i.e. Ls ι I F(x ι ) F(x 0 ). We give first three examples which shows that this notion is independent of both lower and upper semicontinuity notions. Example (a) The multifunction F 1 : [0, 1] [0, 1] defined by x F 1 (x) :=, if 0 x x + 1, if 1 x is outer semicontinuous and upper semicontinuous at x 0 = 1/2 but is not lower semicontinuous at x 0. (b) The multifunction F 2 : [0, 1] R 2 defined by F 2 (x) := {x} R is lower semicontinuous and outer semicontinuous at each point but is not upper semicontinuous at any point. (c) If B := {(x, y) R 2 : x 2 + y 2 < 1} 65

13 4.2 Outer semicontinuity 66 is the open unit ball in the plane then the constant multifunction F 3 : [0, 1] [0, 1] 2, defined by F 3 (x) := B is lower and upper semicontinuous but is not outer semicontinuous at any point. Under some additional assumptions the outer semicontinuity is quite close to the upper semicontinuity. First we give some equivalent presentations of the local outer semicontinuity. THEOREM For any multifunction F : X Y the following statements are equivalent (i) F is outer semicontinuous at x 0 X, (ii) For each net (x ι ) x 0 and each selection net (y ι F(x ι )) y 0 we have y 0 F(x 0 ), (iii) F(x 0 ) = {F[U] U N(x 0 )}, (iv) For each y 0 Y \ F(x 0 ) there exist (open) neighbourhoods V y 0 and U x 0 such that F[U] V =. Proof. (i) (ii): Let (x ι ) x 0 and consider any convergent selection net (y ι F(x ι )) y 0 Y. (it exists if x ι DomF, ι I, and it is always the case if F is a correspondence). Then y 0 LiF(x ι ) LsF(x ι ) and the assumption (i) gives LsF(x ι ) F(x 0 ) therefore y 0 F(x 0 ). (ii) (i): Let us suppose the negation of (i) would hold. It means that there exists a convergent net (x ι : ι I) x 0 such that LsF(x ι ) F(x 0 ). Then y 0 LsF(x ι ) \ F(x 0 ), ( ) but y 0 LsF(x ι ) cofinal K I (y ι F(x ι )) ι K y 0. The cofinal subnet (x ι : ι K) also converges to x 0 and using the assumption (ii) we must have y 0 F(x 0 ) which contradicts ( ). (ii) (iii): Each neighbourhood U N(x 0 ) contains x 0 so the following inclusions hold true F(x 0 ) {F[U] : U N(x 0 )} {F[U] : U N(x 0 )}. To prove the opposite inclusion let y {F[U] : U N(x 0 )}. We construct convergent nets x ι x 0 and y ι y with y ι F(x ι ) for all ι I in the following way. For any U N(x 0 ) we have y F[U] hence for any neighbourhood of y we have V F[U], 66

14 4.2 Outer semicontinuity 67 and using the axiom of choice we can choose y U,V V F[U], and the corresponding x U,V U with y U,V F(x U,V ). The Cartesian product N(x 0 ) N(y) =: I is directed by the product order It is easy to see that (U, V ) (U, V) U U & V V. (x U,V : (U, V) I) x 0, (y U,V : (U, V) I) y. Using the assumption (ii) we get y F(x 0 ) so the opposite inclusion to ( ) follows. (iii) (ii): Let (x ι : ι I) =: x x 0 and (y ι F(x ι ) : ι I) =: y y 0. To prove that y 0 F(x 0 ) it suffices to prove that each neighbourhood V of y 0 meets F[U] for every neighbourhood U of x 0. From the convergence of x and y we have that the sets R := x 1 [U] and S := y 1 [V] are residual in I so R S and for each ι R S we have V F(x ι ) V F[U], and the conclusion follows. (iii) (iv): It is obvious because the condition in (iv) is just another formulation of the implication y F(x 0 ) y {F[U] : U N(x 0 )}, obviously equivalent with {F[U] : U N(x 0 )} F(x 0 ), which is equivalent with (iii). Corollary If F : X Y is outer semicontinuous at x 0 then F(x 0 ) is a closed subset in Y. The global notion of the outer semicontinuity of a multifunction F : X Y is defined as usually requiring F to be outer semicontinuous at any point of its domain DomF, i.e. F : X Y is an outer semicontinuous multifunction if for each x DomF and each net (x ι, y ι ) GrF converging to a point (x, y) in the product space X Y we have (x, y) GrF, in other words GrF is a closed set in DomF RngF. A direct consequence of this notion of the outer semicontinuity is the following 67

15 4.2 Outer semicontinuity 68 Proposition The multifunction F is outer semicontinuous iff its inverse F 1 is o.s.c. A multifunction with a closed graph need not be upper semicontinuous. Example 4.21 shows that also the converse need not be true. But under some mild conditions upper semicontinuity for multifunctions having closed values gives outer semicontinuity. THEOREM Let X and Y be Hausdorff spaces and multifunction F : X Y be u.s.c. at x 0 X and F(x 0 ) be closed. Then (i) if Y is regular then F is o.s.c. at x 0, (ii) if both X and Y are first countable then F is o.s.c. at x 0. Proof. For (i), let y Y \ F(x 0 ) be arbitrary and by regularity let B be a closed neighbourhood of y disjoint from F(x 0 ). By upper semicontinuity, there is a neighbourhood U of x 0 with F[U] B c. As a result, F[U] B c =, so by Theorem 4.23 (iv), F is o.s.c. at x 0. For (ii), suppose F is not o.s.c. at x 0. Then there exists y 0 F(x 0 ) c such that each neighbourhood of y 0 hits F[U] for each U N(x 0 ). Let {U n : n N} and {V n : n N} be countable local bases at x 0 and y 0, respectively, where for each n, U n U n+1, V n V n+1, and V n F(x 0 ) =. Choose for each n N a point x n U n for which F(x n ) V n, and then choose y n F(x n ) V n. Let B := {y n : n N} c, which by the convergence of y n y 0 is an open subset of Y. We have F(x 0 ) B whereas for each n, F(x n ) B. This violates u.s.c. of F at x 0. Outer semicontinuity gives upper semicontinuity with sufficient compactness. THEOREM Let X, Y be Hausdorff spaces. If F : X Y is o.s.c. at x 0 and there exists a neighbourhood U 0 of x 0 with clf[u 0 ] compact, then F is u.s.c. at x 0. Proof. Let K := clf(x 0 ) and let V be an open neighbourhood of F(x 0 ). If V contains K, then F[U 0 ] V and we are done. Otherwise, set K = K \ V, a nonempty compact set. By outer semicontinuity, for each y K there exist neighbourhoods U y of x 0 and V y of y such that F[U y ] V y =. Choose by compactness {y 1, y 2,..., y n } K such that K n i=1 V y i. Then F maps the neighbourhood U 0 U y1 U yn into V, and so F is u.s.c. at x 0. From the above two theorems we can see that globally outer semicontinuous multifunctions are pretty close to globally upper semicontinuous multifunctions with nonempty compact values and it is common to call the latter usco maps. The other types of continuities of multifunctions are defined by means of semicontinuous single-valued functions so we are going to study such functions in the next section. 68

16 4.3 Semicontinuous functions Semicontinuous functions Semicontinuous functions were introduced at the beginning of 20th century by Baire. They share many good properties of continuous functions and behave more friendly with respect to some limit operations and play an important role in optimization. So historically the notion of semicontinuity for functions precedes the similar one for multifunctions and it is not identical with it despite the same name. We start with reminding you of a basic characterization of a continuous real function f defined on a topological space X. Such a function is continuous iff for each a R the sets {x X f (x) > a} (l) {x X f (x) < a} (u) are open in X. It is due to the fact that the unbounded intervals (a, + ) and (, a), (a R) form a subbase of the Euclidean topology on R. If only one of the conditions (l) or (u) is satisfied, we arrive at the notion of semicontinuous functions. Semicontinuous functions behave well with respect to taking infima and suprema but these operations often result in infinite values. So we will introduce semicontinuity for extended realvalued functions, i.e. functions taking values in the extended real line ˆR := [, + ] considered with the natural linear order. Instead of topology induced by this linear order we will consider two coarser topologies on ˆR namely the lower topology: τ l := {(a, + ] a ˆR} { ˆR}, and the upper topology τ u := {[, a) a ˆR} { ˆR}. Unfortunately, these topologies are not Hausdorff (i.e. they do not separate points) but satisfy T 0 separation axiom, i.e. for each two distinct points a b ˆR there exists an open set containing one of them and not containing the other one. The union τ l τ u is a subbase of the usual order topology τ o on ˆR which is obviously Hausdorff. Now we can introduce the local version of semicontinuity of extended real-valued functions. Definition Let X be a topological space. A function f : X ˆR is called lower [upper] semicontinuous at a point x 0 X, abbreviated l.s.c. at x 0 or just f is lsc(x 0 ), if for each a R with f (x 0 ) > a [ f (x 0 ) < a] there exists a neighbourhood U of x 0 such that the following implication holds true x U f (x) > a, (l) [ x U ] f (x) < a. (u) 69

17 4.3 Semicontinuous functions 70 We say that f is lower [upper] semicontinuous on a set M X if it is such at each point of the set M. We say that f is lower [upper] semicontinuous, abbreviated l.s.c. [u.s.c.] if it is such at every point of its domain X. We can express the implications (l) and (u) in the equivalent way, namely for each a < f (x 0 ) [a > f (x 0 )] the set f 1 [(a, + ]] [ f 1 [[, a)]]] is a neighbourhood of x 0. Example (a) The function f : R ˆR defined by f (x) = { 1 if x 0, 1 if x > 0, is l.s.c. but is not u.s.c. at x = 0. (b) The function g : R ˆR defined by is u.s.c. but is not l.s.c. at x 0 = 0. g(x) = { 1 if x < 0, 1 if x 0, The following proposition shows that semicontinuity is closely related to local extrema. Proposition Let f : X ˆR has a local minimum [maximum] at a point x 0 X. Then f is lower [upper] semicontinuous at x 0. Consequently, if f (x 0 ) = [ f (x 0 ) = + ] then f is lower [upper] semicontinuous at x 0. Proof. If f has a local minimum at x 0 then there exists a neighbourhood U of x 0 satisfying the implication x U f (x) f (x 0 ). So for each a < f (x 0 ) we have x U f (x) > a, which is the defining implication of the lower semicontinuity at x 0. A similar proof is valid for the upper semicontinuity. There are number of simple characterizations of semicontinuous functions. The following proposition summarizes some basic characterizations of globally semicontinuous functions but the appropriate localization of these properties is also valid. We leave the proof of the properties for an exercise. Proposition For any function f : X ˆR the following assertions hold true (i) f is l.s.c. f is u.s.c. 70

18 4.3 Semicontinuous functions 71 (ii) f is l.s.c. f is continuous with respect to the lower τ l topology on ˆR, i.e. a R : f 1 [(a, + ]] is open in X (iii) f is u.s.c. f is continuous with respect to the upper τ u topology on ˆR, i.e. a R : f 1 [[, a)] is open in X (iv) f is l.s.c. a R : f 1 [[, a]] is closed in X, the set f 1 [[, a]] is called a sublevel set for f at height a and is denoted slv( f, a) (v) f is u.s.c. a R : f 1 [[a, + ]] is closed in X, the set f 1 [[, a]] is called an upper level set for f at height a (vi) f is l.s.c. and u.s.c. f is continuous with respect to the order topology τ o on ˆR with a subbase τ l τ u Because of the assertion 4.31(i) we can focus on lower semicontinuous functions and by dualisation of our results we obtain valid properties for upper semicontinuous functions. Very convenient visualization of semicontinuous functions is by means of an epigraph or a hypograph. Definition For any extended real-valued function f : X ˆR the epigraph of f [hypograph of f ] is defined by epi f := {(x, λ) X R λ f (x)}, [ hypo f := {(x, λ) X R λ f (x)} ], and it is a subset of X R (not X ˆR). The epigraph and hypograph of a simple discontinuous function are illustrated in Figure 4.3. hypof y = f(x) epif Figure 4.3: Graphic representation of the graph, epigraph and hypograph of the function f. It is evident that a subset E X R is an epigraph of a function iff: (i) E recedes in the vertical direction: (x, α) E, α < β R (x, β) E 71

19 4.3 Semicontinuous functions 72 (ii) E is vertically closed: for each x X, {α R (x, α) E} is closed. Now we can prove the following characterization of globally semicontinuous functions. Proposition The function f : X ˆR is (i) l.s.c. epi f is a closed subset of X R, (ii) u.s.c. hypo f is a closed subset of X R. Proof. We prove (i) and (ii) can be proved when passing to f. If epi f is a closed subset of X R then the sublevel set for f at height a R is the projection on the space X: f 1 [[, a]] = pr X [epi f X {a}]. Since X {a} is closed in X R and the projection pr X : (x, a) x maps a closed set on a closed set, f is l.s.c. because of 4.31(iv). Suppose (x ι, α ι ) ι I is a net in epi f convergent to (x, α) X R in the product topology. To show (x, α) belongs to epi f, we must show α f (x). If f (x) = the claim is obviously valid. Otherwise, take β < f (x) and then by lower semicontinuity of f at x there exists a neighbourhood U of x such that for each y U we have β < f (y). In particular, β < f (x ι ) for ι sufficiently large. Since f (x ι ) α ι we have for each β < f (x): β lim inf ι I f (x ι ) lim ι I α ι = α, hence f (x) α, completing the proof of the assertion (i). From the previous Proposition it is clear that the discontinuous function illustrated in Figure 4.3 is neither l.s.c. nor u.s.c. Next Proposition gives the characterization of local semicontinuity of functions. Proposition Let f : X ˆR and x 0 X. Then (i) f is lsc(x 0 ) for each net (x ι ) ι I x 0 in X then lim inf ι I f (x ι ) f (x 0 ), (ii) f is usc(x 0 ) whenever x ι x 0 in X then lim sup ι I f (x ι ) f (x 0 ). 72

20 4.3 Semicontinuous functions 73 Proof. We prove (ii), the proof of (i) being similar. If f (x 0 ) = + the claim is obviously valid. Let f (x 0 ) < +. Then b > f (x 0 ) U N(x 0 ) x U : b > f (x). To prove the demanded inequality we use the definition of limes superior of a net lim sup ι I f (x ι ) := Inf λ I ( Supι λ f (x ι ) ) = Inf λ I b λ, where b λ := Sup ι λ f (x ι ), for each λ I. So for λ ι 0 we have b λ b and therefore Inf λ I b λ Inf λ ι0 b λ b. Hence lim sup ι I f (x ι ) b is true for any b > f (x 0 ) and consequently lim sup ι I Suppose f were not usc(x 0 ). It means f (x ι ) f (x 0 ). b > f (x 0 ) U N(x 0 ) x U U : b f (x U ). The neighbourhood system N(x 0 ) is upper (left) directed by the inclusion and it is obvious that the net (x U U N(x 0 )) x 0. Then we compute b > f (x 0 ) lim sup U N(x 0 ) ( f (x U ) = Inf U N(x0 ) SupV U f (x V ) ) b. This contradiction shows that f is usc(x 0 ). The following one is a closely related characterization of the local semicontinuity. Proposition For any function f : X ˆR and x 0 X the following assertions are true: (i) f is lsc(x 0 ) lim inf x x0 f (x) = f (x 0 ), (ii) f is usc(x 0 ) lim sup x x0 f (x) = f (x 0 ). 73

21 4.3 Semicontinuous functions 74 Proof. We show the validity of (i) leaving the proof of (ii) as an exercise. First we remind the definition of limes inferior of a function defined on a topological space X with values in ˆR. We consider the neighbourhood system N(x 0 ) upper (left) directed by inclusion. For each N N(x 0 ) there exists Inf f [N] =: a N ˆR. The net (a N : N N(x 0 ), ) is non-decreasing (weakly increasing) so there exists its limit in ˆR and we have: lim N N(x 0 ) a N = Sup N N(x0 ) a N, and we put lim inf := lim (Inf x N f (x)) = Sup N N(x0 ) (Inf f [N]). x x 0 N N(x 0 ) a < f (x 0 ) U N(x 0 ) x U : a < f (x). So a Inf f [U] and also N U : a Inf f [N], and hence So for every a < f (x 0 ) and therefore lim inf x x 0 f (x) := Sup N N(x0 ) a N Sup N U a N a. lim inf x x 0 f (x) a lim inf x x 0 f (x) f (x 0 ). The opposite inequality is also true since for each N N(x 0 ) it is x 0 N, so f (x 0 ) a N and hence f (x 0 ) Sup N N(x0 )a N = lim inf x x 0 f (x). If a < f (x 0 ) = Sup N N(x0 ) a N, then a is an upper bound, so N N(x 0 ) : a < a N := Inf f [N], and hence x N : a < f (x). Proposition For any function f : X ˆR the function defined on X by [ ] φ : x lim inf f (y) ψ : x lim sup f (y) y x y x is l.s.c. [u.s.c.]. 74

22 4.3 Semicontinuous functions 75 Proof. For any x 0 X and any a < φ(x 0 ) := Sup(Inf f [N x0 ]) there exists a neighbourhood U 0 of x 0 with a < Inf f [U 0 ]. Let us consider x U 0 and all N x U 0. Then a < Inf f [U 0 ] Inf f [N x ]. Hence for all N x U 0 we have a < Inf f [U 0 ] Sup Nx U 0 (Inf) f [N x ] Sup Nx (Inf f [N x ]) = φ(x) for all x U 0 and l.s.c. of φ at x 0 follows. Before considering the preservation of the semicontinuity by some algebraic operations we summarize the properties of arithmetic operations extended from R to ˆR in a usual way. In the following formulae the symbol will denote one of the improper real numbers or +, and is the opposite element to. For any (finite) real number x R we put x + = + x =, x =, + =, is undefined x > 0 x = = x, ( x) = = ( x), 0, 0 are undefined, x 0 x := 1 x, x = 0, 0, ±, ± are undefined A consequence of these properties is that ˆR X is not a vector space (over R) so we must be careful in performing some operations with extended real-valued functions. If f : D f ˆR and g : D g ˆR then f + g : D f +g ˆR, with D f +g = {x D f D g { f (x), g(x)} {, }}. Similarly, for f g, f g, f /g. So in the sequel, we will speak about the semicontinuity of functions defined only on a subset D of a topological space (X, τ). In that case the induced topology τ D := {O D O τ} is understood on D. With this in mind we can now formulate semicontinuity properties of functions resulting from algebraic operations with semicontinuous functions. The proof of the following theorem is straightforward and it is asked to be done in an exercise. THEOREM Let f : D f ˆR and g : D g ˆR be lsc(x 0 ). If x 0 is in the domain of the function resulting from respective operations below, the following is true: (i) f + g is lsc(x 0 ), f is usc(x 0 ), (ii) c 0 c f is lsc(x 0 ), c 0 c f is usc(x 0 ), (iii) f 0, g 0 f g is lsc(x 0 ), 75

23 4.3 Semicontinuous functions 76 (iv) f 0 1 f is usc(x 0) if we put 1 0 = +, f 0 1 f is usc(x 0) if we put 1 0 =. Proof. Exercise. The following theorem generalizes the important Weierstrass theorem about the maximum and the minimum of the continuous function on a compact set. THEOREM Let X be a Hausdorff space and let f : X (, + ] =: `R be lower semicontinuous. Suppose that for some a R the sublevel set for f at height a {x X f (x) a} =: slv( f ; a) is compact and nonempty. Then f has a minimum value, i.e. x 0 X : Inf x X f (x) = f (x 0 ). Proof. Evidently, Inf x X f (x) = Inf f (x) a f (x) = Inf( f 1 [(, a]]). By the semicontinuity of f, for each x X with f (x) < + there exists a neighbourhood V x of x such that for each y V x we have f (y) > f (x) 1. By the compactness of f 1 [(, a]] we can find a finite subset {x 1, x 2,..., x n } of the sublevel set such that Inf{ f (x) f (x) a} Inf i n f (x i ) 1 R. This shows that f is bounded below. So there exists µ := Inf f (x) R. For each n N, choose x n slv( f ; a) with f (x n ) min{a, µ+1/n}. By the compactness of slv( f ; a) the sequence (x n ) has a cluster point x 0. For each n N, {x k k n} slv( f ; µ + 1/n) and by the lower semicontinuity of f sublevel sets are closed so x 0 slv( f ; µ + 1/n). This gives f (x 0 ) µ + 1 n for each natural number n and consequently f (x 0 ) µ. It follows f (x 0 ) = µ, completing the proof of the theorem. Corollary The lower [upper] semicontinuous finite-valued function f on a compact Hausdorff space X takes its infimum [supremum] in R and consequently is bounded below [above] on X. 76

24 4.3 Semicontinuous functions 77 Now we study two important operations on families of semicontinuous functions, infimum and supremum. Given a family of functions we define the infimum of the family by ( f i : X ˆR) i I (inf i I f i )(x) := Inf{ f i (x) i I}, and the supremum of the family by for each x X. (sup i I f i )(x) := Sup{ f i (x) i I}, Proposition If each function f i : X ˆR is lower [upper] semicontinuous (at x 0 X) then [ ] f i =: g inf f i =: f i I sup i I is lower [upper] semicontinuous (at x 0 ). Proof. For arbitrary x 0 X and a R, if a < sup i I f i (x 0 ) = g(x), then there exists i 0 I such that a < f i0 (x 0 ) g(x 0 ). Since f i0 is lsc(x 0 ) there exists a neighbourhood U of x 0 such that x U a < f i0 (x 0 ) g(x), hence g is lsc(x 0 ). Similarly, we could prove the u.s.c. of f at any point x 0 X. The functions f or g from Proposition 4.40 need not be l.s.c. (u.s.c.), see Exercises. But for a finite family it is true. Proposition If ( f 1, f 2,..., f n ) is a finite family of l.s.c. [u.s.c.] functions then is l.s.c. [u.s.c.] function. inf( f 1,..., f n ) =: f [ sup( f1,..., f n ) =: g ] 77

25 4.4 Hemicontinuities for multifunctions 78 Proof. For each x 0 X and a R if a < inf{ f 1 (x 0 ),..., f n (x 0 )} = f (x 0 ) then i n : a < f i (x 0 ). Each f i is lsc(x 0 ) so there exist neighbourhoods U i of x 0 verifying x U i a < f i (x) (i = 1, 2,..., n). Therefore for the neighbourhood U := U 1 U 2 U n of x 0 we have x U a < f ix (x) := inf( f 1 (x),..., f n (x)) = f (x), and l.s.c. of f follows. Similarly, we can prove u.s.c. of g. Corollary (a) If f i : X ˆR, (i I) are continuous functions then sup i f i is l.s.c. and inf i f i is u.s.c. (b) If f n f n+1, (n = 1, 2,... ) is an increasing sequence of l.s.c. functions then the pointwise limit f := lim f n : x lim f n (x) ˆR n n is l.s.c. (c) Given any (extended) real function from X, there is a largest [least] l.s.c. [u.s.c.] function majorised [minorised] by f, namely f := sup{h ˆR X h f } [ f := inf{h ˆR X f h} ], i.e. f f f. f [ f ] is called a lower [upper] envelope of f. 4.4 Hemicontinuities for multifunctions If the target space Y of a multifunction F : X Y has richer structure than the topological one, we can define some more types of "continuity" of multifunctions using special features of the target space. Let us suppose (Y, d) be a metric space. On the power set P(Y) we can define an upper hemimetric topology τ h whose local neighbourhood base at an element A P(Y) consists of sets τ h (A) := {B P(Y) e(b, A) < ɛ, ɛ (0, + ]} 78

26 4.4 Hemicontinuities for multifunctions 79 where e(b, A) := Sup{d(x, A) : x B} is the excess of the set B over the set A. We adopt the convention that if A, then e(, A) = 0. The excess functional e is not symmetric. For example, on the real line with the usual metric e([0, 4], [5, 6]) = 4 and e([5, 6], [0, 4]) = 2. So it makes sense to define on P(Y) the lower hemimetric topology τ h with a local neighbourhood base at any element A P(Y) given by τ h (A) := {B P(Y) e(a, B) < ɛ, ɛ > 0}. Definition Let X be a topological space and (Y, d) be a metric space. The multifunction F : X Y is lower [upper] hemicontinuous if the associated set-valued function ˆF : X P(Y) is continuous with respect to the lower [upper] hemimetric topology τ h [τ h ] on the power set P(Y). The next theorem gives the relationships of the semicontinuity of multifunctions. THEOREM Let X be a topological space and Y a metric space and let F be a correspondence from X to Y. (i) If F is u.s.c., then F is upper hemicontinuous. (ii) If F is lower hemicontinuous, then F is l.s.c. In both cases, the converse is not necessarily true. Proof. Both claims are a consequence of the relationships between upper or lower hemimetric topology and the upper (U) or lower (L) topology on P(Y), namely τ h U or τ h L. The following examples show that the converse implications need not be true. Example (a) Let X and Y both be the real line R with the usual topology. Consider the correspondence F : R R, F(x) := (x 1, x + 1). Then it is clear that x 1 x 2 < ɛ e(f(x 1 ), F(x 2 )) < ɛ so F is upper hemicontinuous. On the other hand, F + [( 1, 1)] = {x R : (x 1, x + 1) ( 1, 1)} = {0} which is not an open subset of R and hence F is not upper semicontinuous. (b) Let X = {0, 1, 1,..., 1,...} and Y = R, both with the usual topology. Let F : 2 n defined by ( ) 1 F n = {0, 1, 2,..., n} F(0) = N X R be 79

27 4.4 Hemicontinuities for multifunctions 80 Then F is l.s.c. since any open set G R which intersects N also intersects F(1/n) for all sufficiently large n. However, F is not lower hemicontinuous at 0 since ( ( )) 1 e F(0), F = + n for all values of n. Under some additional conditions, semicontinuity and hemicontinuity are equivalent. THEOREM Let X be a topological space and Y a metric space and F : X Y. (i) If F is upper hemicontinuous and F(x) is compact for each x X, then F is upper semicontinuous. (ii) If F is l.s.c. and F(x) is totally bounded for each x X, then F is lower hemicontinuous. Proof. (i) It is a consequence of the fact that the upper hemimetric topology coincides on the family K(Y) of all compact sets in Y with the trace of the upper topology on K(Y). (ii) It is true because on totally bounded sets the lower hemimetric topology coincides with the lower topology inherited from P(Y). We remark here that if F is a function, then F(x) is a singleton and hence is compact so that all these notions of continuity coincide for functions. But outer semicontinuity is still different even for functions since being equivalent to closed graph, such a function need not be continuous. Another type of hemicontinuity is defined for multifunctions with values in a normed linear space (Y,, ). First we define a support function. For any linear functional p Y and a set C Y we define σ(p, C) := Sup p[c] and the extended real-valued function σ(, C) : Y [, ] is called a support function of the set C. A convex closed set can be represented by its support function since y C iff p(y) σ(p, C) for all p Y. Obviously σ(p, C) = iff C =. Definition Let X be a topological space and Y a real normed linear space. We shall say that the multifunction F : X Y is (scalarwise) upper hemicontinuous at a point x 0 X if for all p Y the extended real-valued function X x σ(p, F(x)) ˆR, is upper semicontinuous at x 0. It is upper hemicontinuous (abbreviated u.h.c.) if it is u.h.c. at all points x 0 X. 80

28 4.4 Hemicontinuities for multifunctions 81 Remark (1) It is not common to define a scalary lower hemicontinuous function even though formally it would be possible demanding the lower semicontinuity of a convenient real-valued function. (2) We use the adjective "scalary" just to distinguish this type of the continuity of multifunctions from the hemicontinuity defined earlier using Hausdorff hemimetric. Often it is called just an upper hemicontinuous multifunction what we will adopt in this section. First we indicate the link between upper hemicontinuous and upper semicontinuous multifunctions. Proposition Any multifunction upper semicontinuous (at x 0 ) is upper hemicontinuous (at x 0 ). Proof. For fixed ɛ > 0 and p Y, there exists a neighbourhood N(x 0 ) such that x N(x 0 ) : F(x) F(x 0 ) + ɛb where B is an open unit ball in the target space Y of F. Then also x N(x 0 ) we have σ(p, F(x)) σ(p, F(x 0 )) + ɛ p since σ(p, ɛb) = ɛ p. Since ɛ can be chosen arbitrarily small, F is hemicontinuous at x 0. Upper hemicontinuity is weaker than upper semicontinuity but often it is sufficient to ensure the closedness of the graph. Proposition The graph of an upper hemicontinuous multifunction with convex closed values is closed. Proof. Consider a net (x ι, y ι ) Gr(F), (ι I) converging to the pair (x, y). Since the functions x σ(p, F(x)) are upper semicontinuous for all p Y, the inequalities imply, by passing to the limit that p(y) = lim ι I and hence p(y) σ(p, F(x)) which implies that p(y i ) σ(p, F(x i )) p(y ι ) lim sup σ(p, F(x ι )) σ(p, F(x)) ι I y cl(co(f(x))) = F(x). The types of continuities of multifunctions considered in this chapter are the most common in literature. Some other types will be considered in specialized sections of this textbook. 81