16Kuiper, G. P., Comm. Lunar Planetary Lab., University of Arizona, 2, 90 (1964).

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1 1494 MATHEMATICS: G. T. WHYBURN PROC. N. A. S. 12 Tousey, R., "The solar spectrum in space," Astronautics, 6, 32 (July 1961). 1"Steacie, E. W. R., Atomic and Free Radical Reactions, Am. Chem. Soc. Monograph (New York: Reinhold Publishing Corp., 1954), 2nd ed., vols. 1 and Rubey, W. W., Bull. Geol. Soc. Am., 62, 1111 (1951). 51 Kuiper, G. P., private communication. 16Kuiper, G. P., Comm. Lunar Planetary Lab., University of Arizona, 2, 90 (1964). CONTINUITY OF MULTIFUNCTIONS* BY G. T. WHYBURN UNIVERSITY OF VIRGINIA Communicated October 11, Introduction.-The characterization of continuity of a function in terms of its preservation of certain topological properties of its domain or range has been of interest over an extended period of years. In 1926 C. H. Rowe' showed that a real-valued function on an open region (of Euclidean space) is continuous if and only if it has the Darboux property (i.e., preserves connectedness) and has closed point inverses, and he noted that this also holds for functions on closed regions possessing a very restricted form of local arcwise connectedness. In 1927 the author2 showed that these two conditions, preservation of connectedness and closed point inverses, characterize continuity of real functions on a closed set X (in Euclidean space) if and only if X is locally connected. In more recent times numerous studies of this same sort but dealing with functions having much broader types of topological spaces as domain and range have appeared. These include, for example, Klee and Utz,3 Halfar,4 5 Pervin and Levine,6 Tanaka,7 Proizvolov,5 the author,9. 10 and others. All of these have involved preservation of properties such as connectedness, compactness, or closedness under the function or its inverse with varying restrictions such as local connectedness, the Hausdorff property, semilocal connectedness, compactness, weak separability, or the like, imposed on the domain or range space as may be appropriate. In the present paper a study of the subject will be initiated from the viewpoint of multiple-valued functions, so-called multifunctions, rather than ordinary (single-valued) functions. The decided advantage that accrues from this approach stems largely from the symmetry that exists between a multifunction and its inverse, both coming directly under the same set of definitions and theorems. This situation does not hold for ordinary functions since their inverses, in general, are not single-valued. It will be shown that four basic and simple results on maltifunctions can be formulated and very easily proven which yield as direct consequences a large body of theorems concerning continuity of both multifunctions and functions, including nearly all 'those cited earlier. The four assertions, with necessary definitions given below, in section 2, and where X and Y are arbitrary open set topological spaces, are as follows: (A) A multifunction f: X -- Y is compact-valued and usc (upper semicontinuous) if and only if it preserves directedness of families.

2 VOL. 54, 1965 MATHEMATICS: G. T. WHYBURN 1495 (B) If X is locally connected, any connectedness preserving multifunction f : X -> Y for which Y is peripherally f-normal is usc. (C) Any usc multifunction f: X -- Y for which Y is f-regular maps every compact set onto a closed set. (D) Let f: X ==) Y (onto) preserve connectedness and have closed, nonmingled point values. Then, if the inverse of a closed image set in Y is connected, it is also closed. Remark: In interpreting these and other statements on multifunctions, it should be kept in mind that upper semicontinuity (use) for a multifunction reduces precisely to continuity in case of single-valuedness when the multifunction beconmes an ordinary function. 2. Preliminaries.-The letters X and Y will denote spaces with open set topologies. All further restrictions will be explicitly stated when needed. The empty set is denoted by (D. For an elementary discussion of multiple-valued functions, which we call multifunctions in contrast to the word function which has come to connote single-valuedness, along with basic definitions and theorems on semicontinuity and continuity of such, the reader is referred to the book by Hahn.I' It will be recalled that such a multifunction f: X -> Y associates with each x e X one or more points of y so that the image of (or value at) x is a set f(x) of points ill Y. For any subset Xo of X, the image f(xo) of Xo is the union of the image sets f(x) for all x E Xo. The inverse f-' of f is likewise a multifunction whose domain is the total image f(x) of X so that we have f-' :f(x) -- X and the inverse (f-1) of f-1 is identical with f. Note that the domain of f-' is f(x) and not necessarily Y unless f is onto. When f is required to be onto, this is indicated by use of the double arrowf: X ==) Y. In general, where f-' is applied to a subset B of Y, f-' (B) means f-' [B.f(X) ]. A multifunction f: X -- Y is upper (lower) semicontinuous, designated use (lsc), provided the inverse of every set closed (open) in f(x) is closed (open) in X. Note that, as remarked earlier, either type of semicontinuity reduces to ordinary continuity in case of a (single-valued) function. In this paper we will deal exclusively with upper semicontinuity. A multifunction f:x -* Y is said to have nonmingled point images provided that for x,,x2 e X with x i* x2, the image sets f(x,) and f(x2) are either disjoint or identical. It is easily checked that if f has nonmingled point images, so also has f'1. Further, f has nonmingled point values if and only if ff-'f - f on X or, equivalently, (a) f-iff-i = f-1 on f(x), or (b) for each image set B c f(x), ff-'(b) = B, or (c) for each inverse set A c X, f-lf(a) = A. A nonempty collection of nonempty sets 5f in a topological space is called a directed family, provided the intersection F, -F2 of any two elements F, and F2 of if contains some element F3 of W. A directed family if is said to cluster at a point p, and p is then a cluster point of 9i, provided every open set Al containing p intersects each element F of i. Also, if converges to a point p provided every open set cu about p contains some element of Wf. A directed family i' will be called a directed under family of a directed family 5, provided every element F of if contains a subset F' which is an element of i'. Finally, a directed family 9i will be said to be directed toward a set E, provided every directed under family of if has a cluster point in E. It should be noted that no directed family can be directed toward the empty set. Now if f: X -- Y is any multifunction, we readily verify the following assertions:

3 1496 MATHEMATICS: G. T. WHYBURN PROC. N. A. S. (a) For any directed family 9 in X, DI = f(m) = Uj(M) ],M e 9E, is a directed family in Y. (a') The same for 91 in f(x) under f- :f(x) X. (b) If 1 is a directed family in X and 9' is any directed under family of f(m) = 91, then 911 andf-' (91') = 9' have a common directed under family 91 = [A1-lM'],M E 911,M' E 11'. We note first that every such set MA Al' is nonempty. For f(m) contains an element N,' of N1' and Al' = f-l (N2') for some N2' e 91'. Thus if N' c N,' -N2', N' e 9', Al' D f-' (N') and Muf-l (N') i 4) so that M M' * (D. Also for Ml" = - Ml * Ml', M2' = M2*M2', we have Ml" M2" = (Ml * Ml') * (l2 * M2') = (MI * M2)I (Ml'-M2') D M3,M3' E 9n", where M3 c Ml M2, M,' c Ml' M2'; and any M 911 and any M' e 9' each contain M -M' e 1" so that 1" is an under family of both 9 and 9'. We shall need also the following easily verified: LEMMA 1. Given a multifunction f:x -* Y and a set A c X. If for each a e A and each directed family 1 in X directed toward a, [f(m) ], Al e 9, is directed toward f(a), then for each family 8 in X directed toward A, 5f = [f(e)], E e 8, is directed toward B = f(a). For suppose, on the contrary, that some directed under family 91 of 5F has no cluster point in B. There exists a directed family 1 in X which is an under family both of 8 and of [f-1(n)], N E 91. Since 8 is directed toward A, some directed under family 1' of 911 converges to a point a e A. Again, since 91' is a directed under family of [f-l(n)], N e X1, there exists a common directed under family 91' of [f(m')], M' E 9', and 91. However, by hypothesis 91' has a cluster point in f(a), whereas 91 has no cluster point in B = f(a) D f(a). A set M is semilocally connected (slc) about a set A c M, provided that for every open set U containing A there exists an open set V with A c V C U such that M - V has only a finite number of components. If M is slc about each of its points, it is semilocally connected (see the author's paper'2). We shall need the following result relating this property to that of local connectedness. LEMMA 2. A peripherally compact connected Hausdorff space X is slc about each of its compact sets if and only if it is locally connected. Suppose first that X is slc about each of its compact subsets. Let p E X and let U be an open set about p. By peripheral compactness we may suppose the boundary C of U to be compact. Since X is slc about C and is Hausdorff, there exists an open set V satisfying C c V c V7 c X - p and such that X - V has only finitely many components. Thus if N is the component of X - V containing p, we have p e N C U and p is not a limit point of X - N. Hence, X is locally connected at p. Now suppose X is locally connected. Let K be any compact set in X and U be any open set about K. Again we may suppose the boundary F of U is compact. Then, since X is Hausdorff, there exist disjoint open sets R and S containing K and F, respectively. If V is an open set about K with compact boundary E and contained in the open set U. R, we have V c U. Now, since the components of X- V are open and cover F, only finitely many of them can intersect F, say F C C, + C Cn. If N1 is the component of X - V containing Cf, i = 1,2,..., n, we have X - U c N = ENo, since X is connected. Thus, W =

4 VOL. 54, 1965 MATHEMATICS: G. T. WHYBURN 1497 X - N is an open set about K contained in U with complement N having finitely many components. The range space Y of a multifunction f: X -) Y is said to be f-regular provided each point image P is separable in Y from each point q of its complement by disjoint open sets, i.e., P and q lie in disjoint open sets; and Y is peripherally f-normal, provided each point image P is separable in Y from each closed set C in its complement by a set K having a closed inverse underf, i.e., there is a separation Y-K = Yp + Y, with P c Yp, Cc Yc. 3. THEOREM A. A multifunction f: X -> Y is compact-valued and USC if and only if it preserves directedness of families. Proof: Suppose first that f is compact-valued and usc. To prove that f preserves directedness of families, by section 2 it suffices to show that for each directed family 9 in X converging to a point x e X, the image family 91 = [f(m) ], M e 911 is directed toward the set f(x). To prove this latter statement, let 91, be any directed under family of 91. Then 9 and the family 911' = [f-1(n1) ], N1 e 91, have a common under family St and on' converges to x. Now suppose 91, has no cluster point in f(x). Then for each y e f(x) there exists Since f(x) is compact, an open set U, about y and N, e 911 such that N1. U, = 4b. n it is covered by a finite union U = E Uv, of the sets U,. The corresponding intersection N1, N,2... Nan contains an element NL1 of 9Z1 and N1* U = 4. However, since f is use, f'1 (Y - U) is closed so that X - f- (Y - U) is open and contains x but does not intersect the element f-1 (N1) of 9'. Thus the under family 9' of 91' cannot converge to x and we have a contradiction. Hence 9i has a cluster point in f(x) and 91 is directed toward f(x). Now assume that f preserves directedness of families. Then for any x e X, [x ] is a directed family in X directed toward x. Hence, the family Vf(x) ] is directed toward the set f(x). Accordingly, every directed family in f(x) has a cluster point in f(x) so that f(x) is compact. To prove that f is use, let B be any closed set in Y, set f' (B) = A, and suppose there is a point x e A - A. Let 9 be the directed family of sets U -A for all open sets U in X containing x. Then M9 converges to x and thus the family 91 = [f(m) ], M e 91, is directed toward the set f(x). However, f(x) lies wholly in the open set Y - B and Y - - B intersects no one of the sets f(m) B for M e 911. This is impossible because it is readily verified that [f(m) B], M e 91, is a directed under family of 91, and hence must have a cluster point in f(x). COROLLARIES. A1. A multifunction f: X =) Y is closed and has compact point inverses if and only if f-1 preserves directedness of families. [(The theorem applied to f'1) (see ref. 10). ] A2. Any usc compact-valued multifunction preserves compactness of sets. (This includes Hahn, ref. 11, 21-34, p. 151.) A3. If a multifunction f is closed and has compact point inverses, the inverse of every compact set is compact. (This is A2 applied to f'1.) A4. A single-valued function is continuous if and only if it preserves directedness of families. 4. THEOREM B. If X is locally connected, any connectedness-preserving multifunction f : X -- Y for which Y is peripherally f-normal is usc.

5 1498 MATHEMATICS: G. T. WHYBURN PROC. N. A. S. Proof: Let B be any closed set in Y and set A = f'l (B). We have to show that A is closed. Suppose, on the (contrary, that there exists x E A -A. Then f(x) C Y - B, and by hypothesis there exists a set C in Y - B - f(x) separating f(x) and B in Y and such thatf-' (C) is closed. Since X is locally connected, the component Q of X - f'1 (C) containing x is opeii. Also, Q -A = 4) because f(q) is connected and contains f(x) but does not intersect C and thus does not intersect B. Hence, x cannot be a limit point of A contrary to our supposition. COROLLARIES. B1. If X is locally connected and Y is Hausdorff and regular in the Menger-Urysohn sense, any connectedness-preserving compact-valued multifunction f : X - Y having closed point inverses is usc. B2. With X and Y as in B1, a single-valued function f: X -- Y is continuous if and only if it preserves connectedness and has closed point inverses. B3. If X is locally connected, a real-valued function on X is continuous if and only if it preserves connectedness and has closed point inverses. B4. If X and Y are Hausdorff, X is locally connected and Y is peripherally compact, a compact-valued compact multifunction f: X -) Y is usc and has connected values if and only if it preserves connectedness. Note: That f is compact means that f-1 preserves compactness. To get B4, recall that a peripherally compact Hausdorff space is regular. 5. THEOREM C. Any usc multifunction f : X -- Y for which Y is f-regular maps every compact set onto a closed set. Proof: Let K be any compact set in X and suppose its image f(k) = H is not closed so that there exists q e 12 - H. Let 9Z be the directed family of all sets [V. H] for all open sets V in Y containing q; and let N be the directed family [K f-'(n)], N e 9l. Since K is compact, a directed under family I' of M1Z converges to some p e K. Now the set P = f(p) does not contain q because q is not in H. By f-regularity of Y, there exists an open set V about q such that V7- P = 4). By usc of f, f-1 (V) is closed and does not contain p. Thus U = X - f-1 (V) is open and contains p. But then U contains some M' e M1Z'; and this is not possible, since U fails to intersect the element K f-1 (V-H) of M. COROLLARIES. C1. If Y is a regular T1 space, any usc multifunction f : X --- Y with closed point images maps every compact set onto a closed set. C2. With Y and f as in C1, if X is compact, f is closed; and thus f-' is usc. C3. If X is regular T1 and g: X -* Y is a closed multifunction with closed point inverses, the inverse of every compact set is closed. (This is C2 applied to f-1.) C4. If f: X -. Y is compactness preserving and has closed point inverses where X and Y are Hausdorff spaces, the inverse of every closed set is compactly closed (i.e., its intersection with every compact set is compact) (see ref. 10). For, taking a closed set F in Y, let E = f-'(f) and let H be any compact set in X. We must show that the set A = E H is closed. Let K = f(h), B = F-K anddefinef H = g. Theng:H=H)K,g-'(B) = A,HandKarenormalT1. Also g is closed (since f is compactness preserving) and has closed point inverses [since for y 6 K, g1-(y) = H.f-1(y)]. Accordingly, by C3, A must be closed. C5. If X and Y are Hausdorff spaces and all compactly closed sets in X are closed, any compactness-preserving multifunction f: X -- Y with closed point inverses is usc. (Thus for such a domain and range, a compact-valued multifunction is usc if and only if it preserves compactness and has closed point inverses.)

6 VOL. 54, 1965 MATHEMATICS: G. 7'. WHYBURN 1499 C5'. If X and Y are Hausdorff spaces and compactly closed sets in Y are closed, any compact multifunction with closed point values g:x =) Y is closed. (This is C5 applied to f = -1.) C6. If X and Y are Hausdorff spaces and X is weakly separable and locally connected, any compactness and connectedness-preserving multifunction f:x -- Y with nonmingled values is usc. This follows from C5, since in this case point inverses are closed. To see this, suppose some point p not in f-'(q) is a limit point of f-l(q) for some q e Y. Then P = f(p) and Q = ff-1(q) are disjoint (by nonmingled condition) and compact. Thus there exists an open set V in Y with Q cvvc 2c Y-P. Now if U1D U2 D U3... is a countable basis of connected open sets at p, sincef(ul) is connected and U1 -f-'(q) * (P, there is a point pi e U1 such that f(pi) (V - Q) * 4. Next, taking an open set V1 in Y with Q c V1 C V1 c Vf- f(p1), we find similarly P2 6 U2 with f(p2) *(V1 - - Q) t 4, and so on indefinitely. This gives a sequence Pn P so that K = p + An is compact. Let ft K = g. Then g preserves compactness and g-1 is single-valued. Thus, g is use by C5. This is impossible, since if E = g(k)- 'V, E is closed, whereas g-i(e) = K - p and this set is not closed. 6. THEOREM D. Suppose f: X Y preserves connectedness and has closed nonmingled point values. Then if the inverse A of a closed image set B c Y is connected, it is closed. Proof: If there existed an x e A - A, A + x would be connected. However, we would have f(a + x) = f(a) + f(x) = B + f(x), since B = ff-'(b) = f(a) by the nonmingled condition. This is a separation, as B and f(x) are closed and disjoint; and this contradicts the fact that f(a + x) is connected. COROLLARIES. D1. Given f as in Theorem D. If for a set B c Y, A = f-'(b) is connected and f(a) is closed, then A is closed. (For A = f-lf(a) by the nonmingled condition. Thus, f(a) is a closed image set.) D2. With f as in Theorem D, if a point inverse f-'(y) is connected, it is closed. (For ff-1(y) = f(x) for any x e f-'(y), and thus by D1 this set is closed.) D3. Given f:x -- Y single-valued and connectedness-preserving, where Y is T1. If the inverse of a closed set B c Y is connected, it is closed. Thus if f is weakly monotone, it has closed point inverses. D4. Any single-valued doubly connectedness-preserving function f:x ==) Y onto a semilocally connected T1 space Y is continuous. For if B is any closed set in Y, A = f-1(b) and p is any point of X - A, semilocal m connectedness of Y at q = f(p) yields a finite union N = NjNf of closed connected subsets of Y - q containing B. Since each f-'(ni) is connected, by D3 f-1(n) is closed; and since this set contains A but not p, A must be closed. D5. If X and Y are semilocally connected Ti-spaces, any 1-1 doubly connectednesspreserving function f: X ==) Y is a homeomorphism. [(This holds in particular, in case X and Y are Euclidean spaces) (see refs. 6 and 7). ] D6. Any doubly connectedness-preserving multifunction f:x ==) Y with closed nonmingled point values onto a space Y which is semilocally connected about the point values of f is usc. D7. If Y is a peripherally compact, connected, and locally connected Hausdorff

7 1500 MATHEMATICS: G. T. WHYBURN PROC. N. A. S. space, any compact and nonmingled valued, doubly connectedness-preserving multifunction f: X ==) Y is usc. D8. Any doubly connectedness-preserving function f: X =) Y from a connected setx of real numbers to a locally connected space Y is continuous. This can be obtained from D7 by showing Y peripherally compact. However, it is just as easy to prove continuity directly. For let R be any region in Y and let x e f-'(r). If X contains a number a < x, the set A of all such numbers is connected, as is also A + x. Thus, f(a + x) would be connected and hence, f(a) intersects R so that f-1(r) contains an a < x. Similarly, if X contains a number > x, so also does f-'(r). In any case, since f-'(r) is connected, it contains an open subset of X about x. Thus f-'(r) is open in X so that f is continuous. 7. Summary of Results for Functions.-These may be conveniently tabulated as follows. TABLE 1 CASES IN WHICH A FUNCTION f: X -- Y IS CONTINUOUS Follows directly Refer- X Y f from ence 1. Open region of E. Real numbers Connectedness-preserving, B3 1 closed point inverses 2. Closed loc. conn. Reals Connec. pres. closed pt. in- B3 2 set in En verses 3. Metric Metric Compactness pres., closed pt. C5 3 inverses 4. Metric, loc. conn. Metric Connec. and compact pres. C6 5. Metric, loc. conn. Metric Connec. pres., Y periph. f- B 3 normal 6. Metric, loc. conn. Reals Connec. pres., closed pt. inv. B Regular T, Compact T, Closed, closed pt. inv. C Reg. Hausdorff Cpt. Hsdf. Closed, w. monotone, connec. C3, D3 5,11 pres. 9. Loc. cpt. Hsdf. Hsdf. Cpt. pres., closed pt. inv. C Hsdf. slc Hsdf. 1-1, Doubly conn. pres., onto D4 6, 11. Cpt. loc. conn. Cpt. 1. conn. 1-1, Doubly conn. pres., onto D, 6, 7 Hsdf. Hsdf. 12. Property K Hsdf. Cpt. pres., closed pt. inv. C Reals Conn., periph. 1-1, Closed, onto B4* 8, 9 cpt. Hsdf. 14. Reals Conn., loc. 1-1, Closed, onto Dg* 8, 9 conn. Additionally, a multifunction f: X=) Y is USC in the cases: Conn. Hsdf. Periph. cpt. Connec. pres., f' a monotone B4 9 Hsdf. mapping 16. Dendrite Conn., periph. f1 a monotone mapping B4* 9 cpt. Hsdf. * Use Theorem 2 of ref. 9. * This research was supported by a grant from the National Science Foundation. 1 Rowe, C. H., "Note on a pair of properties...," Bull. Am. Math. Soc., 32, (1926). 2 Whyburn, G. T., "The most general closed point set...," Bull. Am. Math. Soc., 33, (1927). 3 Klee, V. L., and W. R. Utz, "Some remarks on continuous transformations," Proc. Am. Math. Soc., 5, (1954). 4Halfar, E., "Compact mappings," Proc. Am. Math. Soc., 8, (1957). 6 Halfar, E., "Conditions implying continuity of functions," Proc. Am. Math. Soc., 11, (1960).

8 VOL. 54, 1965 MATHEMATICS: R. BELLMAN Pervin, W. J., and N. Levine, "Connected mappings of Hausdorff spaces," Proc. Am. Math. Soc., 9, (1958). 7Tanaka, T., "On a family of connected subsets and topology of spaces," J. Math. Soc. Japan, 7 (1955). 8Proizvolov, V. V., Doki. Akad. Nauk SSSR, 151, (1963); English translation, Soviet Math. Dokl., 4, (1963). 9 Whyburn, G. T., "On compactness of mappings," these PROCEEDINGS, 52, (1964). 10 Whyburn, G. T., "Directed families of sets and closedness of functions," these PROCEEDINGS, 54, 688 (1965). 11 Hahn, H., Reelle Funktionen (Leipzig: Akad. Verlagsgess, 1932). 12Whyburn, G. T., "Semi-locally connected sets," Amer. J. Math., 61, (1939). A NEW APPROACH TO THE NUMERICAL SOLUTION OF A CLASS OF LINEAR AND NONLINEAR INTEGRAL EQUATIONS OF FREDHOLM TYPE* BY RICHARD BELLMAN THE RAND CORPORATION, SANTA MONICA, CALIFORNIA Communicated by H. S. Vandiver, October 4, Introduction.-In many scientific investigations the final step is the numerical solution of a linear integral equation of the form (P(u) = g(u) + J k(uv)y(v)dv. (1.1) The purpose of this paper is to present a new approach which appears to possess certain advantages in the case where k(u,v) > 0. In subsequent papers we shall describe applications of the method. The reader familiar with the theory of invariant imbedding, and particularly its applications to radiative transfer,1 2 will understand the motivation for the method. As we shall indicate below, it can be applied with equal ease to various classes of nonlinear integral equations. 2. Imbedding.-Let us consider the function of two variables defined as the solution of the integral equation for x > 0 0 < u < 1. V(u,x) = g(u) + J' [f e-(x-)/k(uv) p(vy)dy]dv, (2.1) Then we can state THEOREM. If k(u,v) 2 0, g(u) 2 0, and (1.1) possesses a unique positive solution (p(u), then (2.1) possesses a unique solution which converges monotonically to p(u) as x-oco. This result is easily established by using the sequence of functions defined in the following way: spo(u,x) = g(u), n (u x) = g(u) + f [f e(z -)/k(lu) fnx(vyy)dy dv (2.2) n> 1.