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Theory of fuzzy limits
Fuzzy Sets and Systems 115 (2000) 433–443
www.elsevier.com/locate/fss
Theory of fuzzy limits Mark Burgin Department of Mathematics, University of California, Los Angeles, CA 90095, USA Received November 1997; received in revised form July 1998
Abstract Theory of limits is the base of the classical mathematical analysis (the calculus). In a similar way, theory of fuzzy limits, presented in this paper, is the base of neoclassical analysis, which is a new branch of fuzzy mathematics and extends possibilities provided by classical analysis. The concepts of a fuzzy limit of a sequence and a fuzzy limit of a c 2000 Elsevier Science B.V. All rights reserved. function are introduced and studied. Keywords: Mathematical analysis; Theory of limits; Limit of a sequence; Limit of a function; Fuzzy set theory; Fuzzy limit of a sequence; Fuzzy limit of a function; Fuzzy set of limits; Measure of convergence
1. Introduction Theory of limits is a corner stone of the classical mathematical analysis (calculus). In a similar way, theory of fuzzy limits, presented in this paper, is the base of the neoclassical analysis, which is a new branch of fuzzy mathematics and extends possibilities provided by the classical analysis [1 – 5]. In the second part of this paper a theory of limits of sequences in metric spaces is developed and speciÿed for number sequences. In the third part a theory of limits is developed for functions. It is necessary to remark that all statements (theorems, lemmas etc.) of the classical theory of limits are simple corollaries of the corresponding statements of the theory of fuzzy limits which is an important part of the neoclassical analysis. The main peculiarity of the neoclassical analysis is that it investigates classical mathematical objects (sequences, series, functions etc.) by means of fuzzy concepts (fuzzy continuity, fuzzy limits etc.). This provides di erent possibilities. First, the range of the classical analysis is essentially extended. Second, many classical results are generalized becoming thus explicit consequences of the corresponding neoclassical statements. Third, some of the classical results are completed. Fourth, many new applications are found based on more adequate models of real phenomena. For example, it is known that any convergent sequence is bounded. The converse is not true. So, this theorem of the classical analysis gives only sucient conditions. A complete criterion is proved in this paper. It is demonstrated (Theorem 2.3) that a sequence (of real numbers or elements of a metric space) is bounded if and only if it fuzzy converges. c 2000 Elsevier Science B.V. All rights reserved. 0165-0114/00/$ - see front matter PII: S 0 1 6 5 - 0 1 1 4 ( 9 8 ) 0 0 3 3 8 - 8
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It is necessary to remark that only some of possibilities of the neoclassical analysis are presented in this paper. Others may be found elsewhere. Denotations. ! is the set of all natural numbers; R is the set of all real numbers; R+ = {r ∈ R; r¿0}; if l = {ai ∈ M ; i ∈ !}, and f : M → L then f(l) = {f(ai ); i ∈ !}.
2. Fuzzy limits of sequences in metric spaces Let L be a metric space with a metric , r be a non-negative real number, and l = {ai ; ai ∈ L; i ∈ !} be a sequence in L. The most natural examples in all cases are given by number sequences. Deÿnition 2.1. An element a ∈ L is called an r-limit of l (it is denoted a = r-limi→∞ ai or a = r-lim l) if for any k ∈ R+ \{0} the inequality (a; ai )6r + k is valid for almost all ai . Remark 2.1. An r-limit may be deÿned for an arbitrary set H ⊆ L as follows: a = r-lim H ⇔ for any k ∈ R+ the inequality (a; x)6r + k is valid for almost all x ∈ H . Lemma 2.1. If a = r-lim l; then a = q-lim l for any q¿r. Lemma 2.2. a = lim l if and only if a = 0-lim l. Deÿnition 2.2. The upper defect (a = lim l) of convergence of l to a is equal to inf {r; a = r-lim l}. Lemma 2.3. If a = lim l and (a; b)¡r; then b = r-lim l. Lemma 2.4. If q = (a = lim l); then a = q-lim l. Proof. Let us consider some k ∈ R+ . By the deÿnition of the inÿnum, there is such a number r ∈ R that r − q¡t = 13 k and a = r-lim l. By Deÿnition 2.1, there is such an m ∈ ! that (a; ai )¡r + t for all i¿m. Thus, (a; ai )¡r + t¡(q + t) + t6q + 2t¡q + k. As k was an arbitrary number, the lemma is proved. Deÿnition 2.3. The upper measure of convergence of l to a point a ∈ L is equal to (a = lim l) =
1 : 1 + (a = lim l)
The upper measure of convergence of l deÿnes the normal [11] fuzzy set Lim l = [L; (x = lim l)] of fuzzy limits of l. Deÿnition 2.4. (a) An element a ∈ L is called a fuzzy limit of a sequence l if (a = lim l)¡∞ or, equivalently, (x = lim l)¿0. (b) A sequence l fuzzy converges if it has a fuzzy limit. Let b ∈ L. Proposition 2.1. If a is a fuzzy limit of l and (a; b)¡∞, then b is the fuzzy limit of l.
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Proposition 2.2. The following conditions are equivalent: (1) a sequence l has no fuzzy limits; (2) some subsequence of l diverges; (3) some subsequence of l has no fuzzy limits; (4) if is ÿnite for any pair of points a; b from L then there is an a ∈ L which is not a fuzzy limit of l; (5) the diameter d({a; a ∈ l}) is inÿnite. Deÿnition 2.5. A sequence h = {bi ; bi ∈ L; i ∈ !} is a subsequence of a sequence l = {ai ; ai ∈ L; i ∈ !} (it is denoted by h ⊆ l) if there is such an injection m : ! → ! that bi = am(i) for all i ∈ !. Lemma 2.5. If h is a subsequence of l and a = r-lim l; then a = r-lim h. Corollary 2.1. If h is a subsequence of l and a = r-lim l; then (x = lim l)¿(x = lim h). Corollary 2.2 [6, Proposition 3:13:10]. If h is a subsequence of l and a = lim l; then a = lim h. Proposition 2.3. If h is a subsequence of l; then any fuzzy limit of l is a fuzzy limit of h, and vice versa when the metric is ÿnite. Theorem 2.1. If a = r-lim l and (a; b) = q¿r; then (ai ; b)¿q − r; for almost all elements ai from l. Proof. If a = r-lim l then by the deÿnition l (ai ; a)6r for almost all ai from l. Then (a; b)6(ai ; b) + (a; ai ). As a consequence, (ai ; b)¿(a; b) − (ai ; a)¿(a; b) − r = q − r. The theorem is proved. Let L = R. Corollary 2.3. If a = r-lim l and a¿b + r; then ai ¿b for almost all ai from l. Corollary 2.4 [6, 10]. If a = lim l and a¿b, then ai ¿b for almost all ai from l. Corollary 2.5 [6, 7, 10]. If a = lim l and a¿0; then ai ¿0 for almost all ai from l. Corollary 2.6. If ai 6q for almost all ai from l and a = r-lim l; then a6q + r. Corollary 2.7 [6, 7, 10]. If ai 6q for almost all ai from l and a = lim l; then a6q. Theorem 2.2. All r-limits of a sequence l belong to some sphere E in the space L the diameter of which is equal to 2r. Proof. If the sequence l has no r-limits then the statement of the theorem is obviously true. Let a = r-lim l be an r-limit of l and b be a point in L for which (a; b)¿2r. Then by Theorem 2.1 (ai ; b)¿2r − r = r for almost all elements ai from l. Thus, b cannot be an r-limit of l. As a consequence, the distance between any pair of r-limits of l is less than or equal to 2r. So, all r-limits of l belong to a sphere with the diameter equal to 2r. The theorem is proved. Corollary 2.8 (any course of mathematical analysis, cf. [6, 7, 10]). A limit of a sequence is unique (if this limit exists).
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Really, a limit of the sequence l is (by Lemma 2.2.) a 0-limit of l. By Theorem 2.2 all 0-limits of l belong to a sphere with the diameter equal to 0. But in metric spaces such a sphere is a single point. Theorem 2.3. A sequence l fuzzy converges if and only if it is bounded. Proof. Suciency. If l is bounded then the distances between any two elements of l are bounded by some number r. Then by Deÿnition 2.1 any element ai from l is an r-limit of a1 . So, l fuzzy converges. Necessity is a consequence of Theorem 2.2. It is possible to introduce inÿnite fuzzy limits for number sequences. Deÿnition 2.6. ∞ (−∞) is an r-limit of l if almost all elements ai are bigger (less) than r. Lemma 2.6. ∞ (−∞) is the limit of l [10] if it is an r-limit of l for any r¿0 (r¡0). In the classical mathematical analysis partial limits (like lim or lim) are also considered. In some problems they play an important role. Thus, in a general case a point a ∈ L is called a partial limit of a sequence l (it is denoted a = lim l) if any neighbourhood of a contains inÿnitely many elements of l. In this context lim l is the biggest and lim l is the smallest partial limit of l. Deÿnition 2.7. An element a ∈ L is called a partial r-limit of a sequence l (it is denoted a = r-plimi→∞ ai ) if for any k ∈ R+ \{0} we have (ai ; a)6r for inÿnitely many elements ai from l. Any partial r-limit of l is also called a fuzzy partial limit of l. Lemma 2.7. If h ⊆ l; then any partial r-limit of h is a partial r-limit of l. Lemma 2.8. Any partial limit is a partial 0-limit and vice versa. Proposition 2.4. The following conditions are equivalent: (1) a sequence l has no partial fuzzy limits; (2) all subsequences of l diverge; (3) all subsequences of l do not have partial fuzzy limits. Let L be a complete metric space. Proposition 2.5. An element a is an r-limit of l if and only if (a; c)6r for any partial limit c of l and l is bounded. Proof. Necessity. Let the point a be an r-limit of a sequence l. Then by Theorem 2.3 the sequence l is bounded. Let c be a partial limit of l and suppose that (a; c) = r + k. Then there is a subsequence h of l for which c = lim h. As a consequence, almost all elements of h belong to the set Ot c = {x ∈ L; (c; x)¡t = 12 k}. Thus, for all such ai ∈ h we have (c; ai )¡t and (a; ai )¿(a; c) − (c; ai ) by the properties of metrics. As (a; c) = r + k, (a; ai )¿r + 12 k. As there are inÿnitely many such ai , it contradicts the statement that the element a is an r-limit of l, and thus, proves the necessity. Suciency. Let l be bounded sequence in L, a ∈ L and (a; c)6r for any partial limit c of l. Suppose that there are inÿnitely many ai in l such that (a; ai )¿r + k for some k ∈ R+ . All these ai constitute a
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subsequence h of l. As L is a complete metric space and h is bounded (because l is bounded), the sequence h has a partial limit d and (a; d)¿r + k. This contradiction to our supposition proves the suciency of the proposition. Corollary 2.9. (a = lim l) = sup{(a; c); c is a partial limit of l}. Remark 2.2. Not all properties of ordinary limits are properties of fuzzy limits. For example, an r-limit is not unique (as aprule) when r¿0. For example, if L = R 2 and l = {(1=i; 1=i); i ∈ !}, then all points of the set U = {(x; y); x 2 + y 2 6r} are r-limits of l for an arbitrary r ∈ R+ . But in some cases an r-limit may be unique as demonstrates the following proposition. Let L be an n-dimensional Euclidean space R n . Proposition 2.6. If q = (l) = inf {r; ∃a = r-lim l} and b = q-lim l then the q-limit of l is unique. Proof. Let Lq (l) be the set of all q-limits of l, a; b ∈ Lq (l), and a 6= b. Then by Theorem 2.3 the interval [a; b] is a subset of Lq (l). Let c be a partial limit of l. Properties of Euclidean spaces imply that there is such a point d ∈ [a; b] for which (d; c)¡max{(a; c); (b; c)}6q. By the deÿnition of q the inequality (d = lim l)¿q. At the same time (cf. Proposition 2.5) (d = lim l) = sup{(d; x); x is a partial limit of l}6q. So, (d = lim l) = q. The space L is complete. So, there is a partial limit e of l for which (d; e) = q. By Proposition 2.5 (a; c)6q and (b; e)6q. But if the ends of an interval [a; b] are inside a sphere (with radius r − k) then all points of [a; b], including d, have to be inside this sphere. It implies (d; e)¡q. This contradiction completes the proof. Remark 2.3. Proposition 2.6 may be invalid if the supposition concerning L is not fulÿlled. It is demonstrated by the following example. Let L be the union of the two closed intervals [−1; 0] and [1; 2] which contain the sequence l = {a2i = −1=i; a2i−1 = 1 + 1=i; i ∈ !}. For this sequence we have (l) = 1 and there are two 1-limits 1 and 0 of the sequence l, i.e. such a limit is not unique. Theorem 2.4. The set Lr (l) = {a ∈ L; a = r-lim l} of all r-limits of a sequence l is convex. Proof. Let a; b ∈ Lr (l) and c ∈ [a; b] where [a; b] is the interval in L which connects a and b. Let us demonstrate that [a; b] ⊆ Lr (l). Really, if (ai ; a)6r + k when i¿n and (ai ; b)6r + k when i¿m, then (ai ; a)6r + k and (ai ; b)6r + k when i¿max{n; m}. The point c is situated on [a; b] between a and b. So, by the properties of Euclidean spaces either (ai ; c)¡(ai ; a) or (ai ; c)¡(ai ; b). In any case (ai ; c)6r +k when i¿max{n; m}. By Deÿnition 2.1 c is an r-limit of l because k is an arbitrary number from R+ . As c is an arbitrary point from [a; b], it is demonstrated that [a; b] ∈ Lr (l). So, Lr (l) is convex [8]. A fuzzy set is convex [11] if all its -levels are convex. Thus, Theorem 2.4 and Deÿnition 2.3 imply the following result. Corollary 2.10. Lim l is a convex fuzzy set.
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Theorem 2.5. Lr (l) is a closed set. Proof. Let h = {bi | bi ∈ Lr (l); i ∈ !} and b = limi→∞ bi . Let us suppose that (b; ai )¿r + k for some k ∈ R+ and inÿnitely many ai from l. These points ai constitute a subset A of l. The metric space L may be isometrically included in a complete metric space M [8]. By Proposition 2.2, A is a bounded set. So, there is a sequence v = {ci ; i ∈ !} ⊆ A which converges to some element c ∈ M . As the inclusion of L in M is isometric (b; c)¿r + k. Then there is bj ∈ Lr (l) such that (bj ; b)¡ 13 k. As a consequence, (bj ; c)¿(b; c) − (b; bj ) = r + 23 k and (bj ; ai )¿r + 13 k for almost all ai from v. This contradicts the supposition that bj ∈ Lr (l) and such a way completes the proof of the theorem. Remark 2.4. Not all essential properties of ordinary limits are valid for fuzzy limits as the following example demonstrates. An important property of limits of number sequences is formulated as follows: if l = {ai ; i ∈ !}, and h = {bi ; i ∈ !}, and ai 6bi for almost all i ∈ ! then lim l6lim h. Let us consider such sequences as l = {1=(i + 1); i ∈ !} and h = {1=i; i ∈ !}. Then 0 is a 1-limit of h and 1 is a 1-limit of l. So, 0 = 1-lim h¡1 = 1-lim l while ai = 1=(i + 1)¡bi = 1=i for all i ∈ !. Deÿnition 2.8. The lower defect d(a = lim l) of convergence of l to a is equal to d(a = lim l) = inf {r | a = r-lim h; h ⊆ l}. Deÿnition 2.9. The lower measure m(a = lim l) = 1=[1 + d(a = lim l)]. Lemma 2.9. m(a = lim l) = 0 ⇔ ∃h ⊆ l (a = lim h) ⇔ a is a limit point (or a partial limit) of l. Proposition 2.7. The following conditions are equivalent: (a) a = lim l; (b) a = 0-lim l; (c) (a = lim l) = l; (d) (a = lim l) = m(a = lim l). Let b ∈ L. Proposition 2.8. (a) (b = lim l)6(a = lim l) + (a; b); (b) d(b = lim l)6d(a = lim l) + (a; b). Deÿnition 2.10. The radius (the measure of divergence) r(l) of the sequence l is equal to r(l) = inf {(a = lim l); a ∈ L}: Lemma 2.10. r(l) = 0 if and only if l converges. Lemma 2.11. t = r(l) if only if for any k ∈ R+ there is a ∈ L such that almost all elements of l belong to the neighbourhood Op a = {c ∈ L; (a; c)¡p = t + k} of the point a. Lemma 2.11 states that r(l) is the radius of the least sphere in L, any spherical neighbourhood of which contains almost all elements from l.
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Deÿnition 2.11. A fuzzy subnumber M is a convex normalized fuzzy set M = (R; : R → I ) such that: (1) sup {(x); x ∈ R} = (a) for exactly one a ∈ R; (2) (x) is piecewise continuous. Deÿnition 2.12 (Zimmermann [11]). A fuzzy number M is a fuzzy subnumber for which sup {(x); x ∈ R} = 1. Proposition 2.9. If L is a complete metric space; then for any sequence l = {ai ; ai ∈ R; i ∈ !} the q-limit of l exists for q = (l). Proof. If such b ∈ L, which is equal to q-lim l, does not exist, then for any z¿q such d¿l exists that d = r-lim l. Let us consider a sequence h = {di ; i ∈ !} for which di = ri -lim l and (ri ; q)¡1=i. The sequence h either has limit points or does not have it. In the ÿrst case let d be a limit point of l. Then d = q-lim l. Really, for any k ∈ R+ there is dj ∈ h for which (dj ; d)¡ 19 k and dj = r-lim l with r −q¡1=k. Then for almost all ai ∈ l we have (ai ; d)6(ai ; dj ) + (dj ; d)¡r + 19 k¡q + 19 k + 19 k¡q + k. As k was chosen arbitrarily, d is a q-limit of l, and in this case the proposition is proved. Rn is a complete metric space. So, when the sequence h does not have limit points, there are such points 1 1 ; p−q¡ 10 . Then their neighbourhoods d; c ∈ h for which (d; c) = t¿2q+1; d = r-lim l; c = p-lim l; r −q¡ 10 1 1 Od = {a ∈ l; (a; d)¡r = 10 } and Oc = {a ∈ l; (a; c)¡p+ 10 } does not intersect, but according to Deÿnition 2.1 each of them has to contain almost all points of the sequence 1. This contradiction completes the proof of Proposition 2.7. Remark 2.5. Proposition 2.9 may be invalid if the supposition concerning L is not fulÿlled. It is demonstrated by the following example. Let L be the union of the two open intervals (−1; 0) and (1; 2) which contain the sequence l = {a2i = −1=i; a2i−1 = 1 + 1=i; i ∈ !}. For this sequence we have (l) = 1 but there are no (in L) 1-limit points of l. Proposition 2.10. If a = r-lim l and (a; b) = p; then b = (r + p)-lim l. Proposition 2.10 implies the following result. Proposition 2.11. (x = lim l) is a continuous function for an arbitrary sequence l. Proposition 2.9 and Deÿnition 2.11 imply the following result. Proposition 2.12. For any sequence l = {ai ; ai ∈ R; i ∈ !} the fuzzy set lim l is a fuzzy subnumber. Proposition 2.10 and Lemma 2.2 imply the following result. Theorem 2.6. For any sequence l = {ai ; ai ∈ R; i ∈ !} the fuzzy set lim l is a fuzzy number if and only if the sequence l converges. Let h = {bi ; bi ∈ L; i ∈ !} be a sequence in L and L be a normed real vector space [9].
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Theorem 2.7. If a = r-lim l and b = q-lim h then (a) a + b = (r + q)-lim (l + h) where l + h = {ai + bi ; i ∈ !}; (b) a − b = (r + q)-lim (l − h) where l − h = {ai − bi ; i ∈ !}; (c) ka = |k| · r-lim (kl) for any k ∈ R where kl = {kai ; i ∈ !}. Proof. (a) By the axiom of a triangle which is valid for any norm [9], we have (ai + bi ; a + b) = k(a + b) − (ai ; bi )k = k(a − ai ) + (b − bi )k6ka − ai k + kb − bi k = (ai ; a) + (b; bi ). So, (ai + bi ; a + b)6(ai ; a) + (b; bi ). As a consequence, (ai + bi ; a + b)6r + q for such indices i that (ai ; a)6r and (b; bi )6q. But these inequalities may be false only for a ÿnite set of indices. Thus, (ai + bi ; a + b)6r + q for almost all i ∈ !, and the ÿrst part of the theorem is proved. (b) The second part of the theorem is proved in a similar way. (c) For any k 6= 0 the conditions (ai ; a)6r and (kai ; ka)6|k| · r are equivalent in normed algebras [9]. Thus, ka = |k| · r-lim (kl) for any k ∈ R, and the theorem is proved. As 0 + 0 = 0 and k · 0 = 0 for any k ∈ R, we obtain Corollary 2.11. (Any course of the calculus, cf. [6, 7, 10]). If a = lim l; b = lim h then (a) a + b = lim (l + h); (b) a − b = lim (l − h); (c) ka = lim (kl). Corollary 2.12. If l; h are positive sequences, then (a + b = lim (l + h)) = (a = lim l) + (b = lim l). Corollary 2.13. If l; h are positive sequences, then (a − b = lim(l − h)) = |(a = lim l) − (b = lim l)|. Corollary 2.14. (ka = lim kl) = |k|(a = lim l). Corollary 2.15. If a is a fuzzy limit of l and b is a fuzzy limit of h; then (a + b) ((a − b), and ka) is a fuzzy limit of l + h (of (l − h), and kl), respectively. By the deÿnition (a + b = lim (l + h)) = =
1 1 = 1 + (a + b = lim (l + h)) 1 + (a = lim l) + (b = lim h) 1 1 − (a = lim l) 1 − (b = lim h) + 1+ (a = lim l) (b = lim h)
because (a = lim l) =
1 − (a = lim l) : (a = lim l)
So (a + b = lim (l + h)) =
(a = lim l) · (b = lim h) (a = lim l) · (b = lim h) + (b = lim h) − (a = lim l) · (b = lim h) + (a = lim l) − (a = lim l) · (b = lim h)
=
1 ; −1 (a = lim l) + −1 (b = lim h) − 1
and we obtain
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Corollary 2.16. (a + b = lim (l + h)) =
1 : −1 (a = lim l) + −1 (b = lim h) − 1
Corollary 2.17. (a − b = lim (l − h)) =
−1 (a = lim l)
1 : − −1 (b = lim h) − 1
Corollary 2.18. (ka = lim kl) =
(a = lim l) : |k| + (1 − |k|) · (a = lim l)
Let L be a normed algebra [9]. Theorem 2.8. If a = r-lim l and b = q-lim h; then ab = p-lim lh where lh = {ai ; bi ; i ∈ !} and p = min{(kak · q + sup kbi k · r); (kbk · r + sup kai k · q)}. 3. Local fuzzy limits of functions Let M; L be metric spaces with metrics which are in both cases denoted by the same letter ; f : M → L be an arbitrary function, and r ∈ R+ . Deÿnition 3.1. An element b ∈ L is called an r-limit of f at a point a ∈ M and denoted b = r-limx→ f(x) if for any sequence l satisfying the condition a = lim l, the equality b = r-lim f(l) is valid. Properties of fuzzy limits of number sequences make it possible to prove the following results. Proposition 3.1. An element b ∈ L is a limit of f(x) at a (in the classical sense [6, 10]) if and only if b = 0-limx→a f(x). Corollary 3.1. A 0-limit of a function is unique (if it exists). Proposition 3.2. If b = r-limx→a f(x); then b = q-limx→a f(x); for any q¿r. Let Lr (f; a) denote the set of all r-limits of the function f at the point a. Proposition 3.3. Lr (f; a) =
T a=lim l
Lr f(l).
The intersection of convex sets is a convex set [8]. Thus, Theorem 2.3 implies Theorem 3.1. Lr (f; a) is a convex set. Deÿnition 3.2. The upper defect (b = limx→a f(x)) of convergence of f(x) to b at a ∈ M is equal to o n inf r; b = lim f(x) : x→a
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Lemma 3.2. If q = (b = limx→a f(x)); then b = q-limx→a f(x). Proof. Let us consider some k ∈ R+ . By the deÿnition of an inÿmum, there is such r ∈ R that r − q¡t = 13 k and b = r-limx→a f(x). By the Deÿnition 3.1, there is such n ∈ ! that a = lim{ai } and (f(ai ); f(a))¡r + t for all i¿m. Thus, (f(ai ); f(a))¡r + t¡(q + t) + t6q + 2t¡q + k. As the number k was arbitrary, we have b = q-limx→a f(x). The lemma is proved. Deÿnition 3.3. The upper measure of convergence of f(x) to b at a ∈ M is equal to b = lim f(x) = x→a
1 : 1 + (b = limx→a f(x))
The measure of convergence of l deÿnes the fuzzy set limx→a f(x) = [L; (y = limx→a f(x))] of fuzzy limits of l. Deÿnition 3.4. (a) An element b ∈ L is called a fuzzy limit of a function f(x) at a point a ∈ L if (b = limx→a f(x))¡∞ or, equivalently, (b = limx→a f(x))¿0. (b) a function f(x) fuzzy converges at a point a ∈ L if it has a fuzzy limit at a point a ∈ L. Proposition 3.4. If a is a fuzzy limit of l and (a; b)¡∞; then b is a fuzzy limit of l. Deÿnition 3.5. An element a ∈ L is called a partial r-limit of f at a point a ∈ M (it is denoted a = rplimi→∞ ai ) if for any sequence l satisfying the condition a = lim l and any k ∈ R+ \{0}, the inequalities (b; f(ai ))6r + k are valid for inÿnitely many elements ai from l. Proposition 3.5. A function f has an r-limit at a point a if and only if f is bounded in some neighbourhood of a. Let L = M = E n . Theorem 3.2. If c = r-limx→a f(x) and d = q-limx→a g(x) then c ± d = (r + q)-limx→a (f ± g)(x). Remark. It is possible to think that c − d = (r − q)-limx→a (f − g)(x). However, it is not the case as the following example demonstrates: Let f(x) = x+1 = g(x); q = r = 1; c = 2; d = 0. Then c−d = 2 = (r+q)-limx→0 (f−g)(x) because f−g = 0, but c − d = 2 6= (r − q)-limx→0 (f − g)(x). Theorem 3.3. If r-limx→a f(x) = c and q ∈ R+ ; then rq-limx→a qf(x) = qc. 4. Conclusion We have demonstrated that in a broader context of fuzzy limits, it is possible to extend and make complete some basic results of the classical mathematical analysis. In the neoclassical analysis many results of classical mathematical analysis are reconsidered in a broader context of fuzzy set theory. This approach does not only bring new results which complete their classical analogues but also produces deeper insights and a better understanding of the classical theory.
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Acknowledgements The author expresses his gratitude to one of the referees for useful suggestions and comments. References [1] [2] [3] [4] [5] [6] [7] [8] [9] [10] [11]
M.S. Burgin, On one theorem of neoclassical analysis, Abstracts AMS (3) (1992) 339. M.S. Burgin, Hypernumbers and fuzzy convergence, Abstracts AMS 15 (1) (1994) 236. M. Burgin, Neoclassical analysis: fuzzy continuity and convergence, Fuzzy Sets and Systems 75 (2) (1995) 291–299. Ä M. Burgin, A. Sostak, Towards the theory of continuity defect and continuity measure for mappings of metric spaces, Matematika, Riga, 1992, pp. 45 – 62. Ä M. Burgin, A. Sostak, Fuzzyÿcation of the theory of continuous functions, Fuzzy Sets and Systems 64 (1) (1994) 71– 81. J. Diedonne, Foundations of Modern Analysis, New York, 1960. G.M. Fihtengoltz, Elements of Mathematical Analysis, GITTL, Moscow, 1955 (in Russian). K. Kuratowski, Topology, New York, 1966. A. Mallious, Topological Algebras, Amsterdam, 1986. Ä G.E. Silov, Mathematical Analysis, Moscow, 1969 (in Russian). K.-J. Zimmermann, Fuzzy Set Theory and its Applications, Boston, 1991.
www.elsevier.com/locate/fss
Theory of fuzzy limits Mark Burgin Department of Mathematics, University of California, Los Angeles, CA 90095, USA Received November 1997; received in revised form July 1998
Abstract Theory of limits is the base of the classical mathematical analysis (the calculus). In a similar way, theory of fuzzy limits, presented in this paper, is the base of neoclassical analysis, which is a new branch of fuzzy mathematics and extends possibilities provided by classical analysis. The concepts of a fuzzy limit of a sequence and a fuzzy limit of a c 2000 Elsevier Science B.V. All rights reserved. function are introduced and studied. Keywords: Mathematical analysis; Theory of limits; Limit of a sequence; Limit of a function; Fuzzy set theory; Fuzzy limit of a sequence; Fuzzy limit of a function; Fuzzy set of limits; Measure of convergence
1. Introduction Theory of limits is a corner stone of the classical mathematical analysis (calculus). In a similar way, theory of fuzzy limits, presented in this paper, is the base of the neoclassical analysis, which is a new branch of fuzzy mathematics and extends possibilities provided by the classical analysis [1 – 5]. In the second part of this paper a theory of limits of sequences in metric spaces is developed and speciÿed for number sequences. In the third part a theory of limits is developed for functions. It is necessary to remark that all statements (theorems, lemmas etc.) of the classical theory of limits are simple corollaries of the corresponding statements of the theory of fuzzy limits which is an important part of the neoclassical analysis. The main peculiarity of the neoclassical analysis is that it investigates classical mathematical objects (sequences, series, functions etc.) by means of fuzzy concepts (fuzzy continuity, fuzzy limits etc.). This provides di erent possibilities. First, the range of the classical analysis is essentially extended. Second, many classical results are generalized becoming thus explicit consequences of the corresponding neoclassical statements. Third, some of the classical results are completed. Fourth, many new applications are found based on more adequate models of real phenomena. For example, it is known that any convergent sequence is bounded. The converse is not true. So, this theorem of the classical analysis gives only sucient conditions. A complete criterion is proved in this paper. It is demonstrated (Theorem 2.3) that a sequence (of real numbers or elements of a metric space) is bounded if and only if it fuzzy converges. c 2000 Elsevier Science B.V. All rights reserved. 0165-0114/00/$ - see front matter PII: S 0 1 6 5 - 0 1 1 4 ( 9 8 ) 0 0 3 3 8 - 8
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It is necessary to remark that only some of possibilities of the neoclassical analysis are presented in this paper. Others may be found elsewhere. Denotations. ! is the set of all natural numbers; R is the set of all real numbers; R+ = {r ∈ R; r¿0}; if l = {ai ∈ M ; i ∈ !}, and f : M → L then f(l) = {f(ai ); i ∈ !}.
2. Fuzzy limits of sequences in metric spaces Let L be a metric space with a metric , r be a non-negative real number, and l = {ai ; ai ∈ L; i ∈ !} be a sequence in L. The most natural examples in all cases are given by number sequences. Deÿnition 2.1. An element a ∈ L is called an r-limit of l (it is denoted a = r-limi→∞ ai or a = r-lim l) if for any k ∈ R+ \{0} the inequality (a; ai )6r + k is valid for almost all ai . Remark 2.1. An r-limit may be deÿned for an arbitrary set H ⊆ L as follows: a = r-lim H ⇔ for any k ∈ R+ the inequality (a; x)6r + k is valid for almost all x ∈ H . Lemma 2.1. If a = r-lim l; then a = q-lim l for any q¿r. Lemma 2.2. a = lim l if and only if a = 0-lim l. Deÿnition 2.2. The upper defect (a = lim l) of convergence of l to a is equal to inf {r; a = r-lim l}. Lemma 2.3. If a = lim l and (a; b)¡r; then b = r-lim l. Lemma 2.4. If q = (a = lim l); then a = q-lim l. Proof. Let us consider some k ∈ R+ . By the deÿnition of the inÿnum, there is such a number r ∈ R that r − q¡t = 13 k and a = r-lim l. By Deÿnition 2.1, there is such an m ∈ ! that (a; ai )¡r + t for all i¿m. Thus, (a; ai )¡r + t¡(q + t) + t6q + 2t¡q + k. As k was an arbitrary number, the lemma is proved. Deÿnition 2.3. The upper measure of convergence of l to a point a ∈ L is equal to (a = lim l) =
1 : 1 + (a = lim l)
The upper measure of convergence of l deÿnes the normal [11] fuzzy set Lim l = [L; (x = lim l)] of fuzzy limits of l. Deÿnition 2.4. (a) An element a ∈ L is called a fuzzy limit of a sequence l if (a = lim l)¡∞ or, equivalently, (x = lim l)¿0. (b) A sequence l fuzzy converges if it has a fuzzy limit. Let b ∈ L. Proposition 2.1. If a is a fuzzy limit of l and (a; b)¡∞, then b is the fuzzy limit of l.
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Proposition 2.2. The following conditions are equivalent: (1) a sequence l has no fuzzy limits; (2) some subsequence of l diverges; (3) some subsequence of l has no fuzzy limits; (4) if is ÿnite for any pair of points a; b from L then there is an a ∈ L which is not a fuzzy limit of l; (5) the diameter d({a; a ∈ l}) is inÿnite. Deÿnition 2.5. A sequence h = {bi ; bi ∈ L; i ∈ !} is a subsequence of a sequence l = {ai ; ai ∈ L; i ∈ !} (it is denoted by h ⊆ l) if there is such an injection m : ! → ! that bi = am(i) for all i ∈ !. Lemma 2.5. If h is a subsequence of l and a = r-lim l; then a = r-lim h. Corollary 2.1. If h is a subsequence of l and a = r-lim l; then (x = lim l)¿(x = lim h). Corollary 2.2 [6, Proposition 3:13:10]. If h is a subsequence of l and a = lim l; then a = lim h. Proposition 2.3. If h is a subsequence of l; then any fuzzy limit of l is a fuzzy limit of h, and vice versa when the metric is ÿnite. Theorem 2.1. If a = r-lim l and (a; b) = q¿r; then (ai ; b)¿q − r; for almost all elements ai from l. Proof. If a = r-lim l then by the deÿnition l (ai ; a)6r for almost all ai from l. Then (a; b)6(ai ; b) + (a; ai ). As a consequence, (ai ; b)¿(a; b) − (ai ; a)¿(a; b) − r = q − r. The theorem is proved. Let L = R. Corollary 2.3. If a = r-lim l and a¿b + r; then ai ¿b for almost all ai from l. Corollary 2.4 [6, 10]. If a = lim l and a¿b, then ai ¿b for almost all ai from l. Corollary 2.5 [6, 7, 10]. If a = lim l and a¿0; then ai ¿0 for almost all ai from l. Corollary 2.6. If ai 6q for almost all ai from l and a = r-lim l; then a6q + r. Corollary 2.7 [6, 7, 10]. If ai 6q for almost all ai from l and a = lim l; then a6q. Theorem 2.2. All r-limits of a sequence l belong to some sphere E in the space L the diameter of which is equal to 2r. Proof. If the sequence l has no r-limits then the statement of the theorem is obviously true. Let a = r-lim l be an r-limit of l and b be a point in L for which (a; b)¿2r. Then by Theorem 2.1 (ai ; b)¿2r − r = r for almost all elements ai from l. Thus, b cannot be an r-limit of l. As a consequence, the distance between any pair of r-limits of l is less than or equal to 2r. So, all r-limits of l belong to a sphere with the diameter equal to 2r. The theorem is proved. Corollary 2.8 (any course of mathematical analysis, cf. [6, 7, 10]). A limit of a sequence is unique (if this limit exists).
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Really, a limit of the sequence l is (by Lemma 2.2.) a 0-limit of l. By Theorem 2.2 all 0-limits of l belong to a sphere with the diameter equal to 0. But in metric spaces such a sphere is a single point. Theorem 2.3. A sequence l fuzzy converges if and only if it is bounded. Proof. Suciency. If l is bounded then the distances between any two elements of l are bounded by some number r. Then by Deÿnition 2.1 any element ai from l is an r-limit of a1 . So, l fuzzy converges. Necessity is a consequence of Theorem 2.2. It is possible to introduce inÿnite fuzzy limits for number sequences. Deÿnition 2.6. ∞ (−∞) is an r-limit of l if almost all elements ai are bigger (less) than r. Lemma 2.6. ∞ (−∞) is the limit of l [10] if it is an r-limit of l for any r¿0 (r¡0). In the classical mathematical analysis partial limits (like lim or lim) are also considered. In some problems they play an important role. Thus, in a general case a point a ∈ L is called a partial limit of a sequence l (it is denoted a = lim l) if any neighbourhood of a contains inÿnitely many elements of l. In this context lim l is the biggest and lim l is the smallest partial limit of l. Deÿnition 2.7. An element a ∈ L is called a partial r-limit of a sequence l (it is denoted a = r-plimi→∞ ai ) if for any k ∈ R+ \{0} we have (ai ; a)6r for inÿnitely many elements ai from l. Any partial r-limit of l is also called a fuzzy partial limit of l. Lemma 2.7. If h ⊆ l; then any partial r-limit of h is a partial r-limit of l. Lemma 2.8. Any partial limit is a partial 0-limit and vice versa. Proposition 2.4. The following conditions are equivalent: (1) a sequence l has no partial fuzzy limits; (2) all subsequences of l diverge; (3) all subsequences of l do not have partial fuzzy limits. Let L be a complete metric space. Proposition 2.5. An element a is an r-limit of l if and only if (a; c)6r for any partial limit c of l and l is bounded. Proof. Necessity. Let the point a be an r-limit of a sequence l. Then by Theorem 2.3 the sequence l is bounded. Let c be a partial limit of l and suppose that (a; c) = r + k. Then there is a subsequence h of l for which c = lim h. As a consequence, almost all elements of h belong to the set Ot c = {x ∈ L; (c; x)¡t = 12 k}. Thus, for all such ai ∈ h we have (c; ai )¡t and (a; ai )¿(a; c) − (c; ai ) by the properties of metrics. As (a; c) = r + k, (a; ai )¿r + 12 k. As there are inÿnitely many such ai , it contradicts the statement that the element a is an r-limit of l, and thus, proves the necessity. Suciency. Let l be bounded sequence in L, a ∈ L and (a; c)6r for any partial limit c of l. Suppose that there are inÿnitely many ai in l such that (a; ai )¿r + k for some k ∈ R+ . All these ai constitute a
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subsequence h of l. As L is a complete metric space and h is bounded (because l is bounded), the sequence h has a partial limit d and (a; d)¿r + k. This contradiction to our supposition proves the suciency of the proposition. Corollary 2.9. (a = lim l) = sup{(a; c); c is a partial limit of l}. Remark 2.2. Not all properties of ordinary limits are properties of fuzzy limits. For example, an r-limit is not unique (as aprule) when r¿0. For example, if L = R 2 and l = {(1=i; 1=i); i ∈ !}, then all points of the set U = {(x; y); x 2 + y 2 6r} are r-limits of l for an arbitrary r ∈ R+ . But in some cases an r-limit may be unique as demonstrates the following proposition. Let L be an n-dimensional Euclidean space R n . Proposition 2.6. If q = (l) = inf {r; ∃a = r-lim l} and b = q-lim l then the q-limit of l is unique. Proof. Let Lq (l) be the set of all q-limits of l, a; b ∈ Lq (l), and a 6= b. Then by Theorem 2.3 the interval [a; b] is a subset of Lq (l). Let c be a partial limit of l. Properties of Euclidean spaces imply that there is such a point d ∈ [a; b] for which (d; c)¡max{(a; c); (b; c)}6q. By the deÿnition of q the inequality (d = lim l)¿q. At the same time (cf. Proposition 2.5) (d = lim l) = sup{(d; x); x is a partial limit of l}6q. So, (d = lim l) = q. The space L is complete. So, there is a partial limit e of l for which (d; e) = q. By Proposition 2.5 (a; c)6q and (b; e)6q. But if the ends of an interval [a; b] are inside a sphere (with radius r − k) then all points of [a; b], including d, have to be inside this sphere. It implies (d; e)¡q. This contradiction completes the proof. Remark 2.3. Proposition 2.6 may be invalid if the supposition concerning L is not fulÿlled. It is demonstrated by the following example. Let L be the union of the two closed intervals [−1; 0] and [1; 2] which contain the sequence l = {a2i = −1=i; a2i−1 = 1 + 1=i; i ∈ !}. For this sequence we have (l) = 1 and there are two 1-limits 1 and 0 of the sequence l, i.e. such a limit is not unique. Theorem 2.4. The set Lr (l) = {a ∈ L; a = r-lim l} of all r-limits of a sequence l is convex. Proof. Let a; b ∈ Lr (l) and c ∈ [a; b] where [a; b] is the interval in L which connects a and b. Let us demonstrate that [a; b] ⊆ Lr (l). Really, if (ai ; a)6r + k when i¿n and (ai ; b)6r + k when i¿m, then (ai ; a)6r + k and (ai ; b)6r + k when i¿max{n; m}. The point c is situated on [a; b] between a and b. So, by the properties of Euclidean spaces either (ai ; c)¡(ai ; a) or (ai ; c)¡(ai ; b). In any case (ai ; c)6r +k when i¿max{n; m}. By Deÿnition 2.1 c is an r-limit of l because k is an arbitrary number from R+ . As c is an arbitrary point from [a; b], it is demonstrated that [a; b] ∈ Lr (l). So, Lr (l) is convex [8]. A fuzzy set is convex [11] if all its -levels are convex. Thus, Theorem 2.4 and Deÿnition 2.3 imply the following result. Corollary 2.10. Lim l is a convex fuzzy set.
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Theorem 2.5. Lr (l) is a closed set. Proof. Let h = {bi | bi ∈ Lr (l); i ∈ !} and b = limi→∞ bi . Let us suppose that (b; ai )¿r + k for some k ∈ R+ and inÿnitely many ai from l. These points ai constitute a subset A of l. The metric space L may be isometrically included in a complete metric space M [8]. By Proposition 2.2, A is a bounded set. So, there is a sequence v = {ci ; i ∈ !} ⊆ A which converges to some element c ∈ M . As the inclusion of L in M is isometric (b; c)¿r + k. Then there is bj ∈ Lr (l) such that (bj ; b)¡ 13 k. As a consequence, (bj ; c)¿(b; c) − (b; bj ) = r + 23 k and (bj ; ai )¿r + 13 k for almost all ai from v. This contradicts the supposition that bj ∈ Lr (l) and such a way completes the proof of the theorem. Remark 2.4. Not all essential properties of ordinary limits are valid for fuzzy limits as the following example demonstrates. An important property of limits of number sequences is formulated as follows: if l = {ai ; i ∈ !}, and h = {bi ; i ∈ !}, and ai 6bi for almost all i ∈ ! then lim l6lim h. Let us consider such sequences as l = {1=(i + 1); i ∈ !} and h = {1=i; i ∈ !}. Then 0 is a 1-limit of h and 1 is a 1-limit of l. So, 0 = 1-lim h¡1 = 1-lim l while ai = 1=(i + 1)¡bi = 1=i for all i ∈ !. Deÿnition 2.8. The lower defect d(a = lim l) of convergence of l to a is equal to d(a = lim l) = inf {r | a = r-lim h; h ⊆ l}. Deÿnition 2.9. The lower measure m(a = lim l) = 1=[1 + d(a = lim l)]. Lemma 2.9. m(a = lim l) = 0 ⇔ ∃h ⊆ l (a = lim h) ⇔ a is a limit point (or a partial limit) of l. Proposition 2.7. The following conditions are equivalent: (a) a = lim l; (b) a = 0-lim l; (c) (a = lim l) = l; (d) (a = lim l) = m(a = lim l). Let b ∈ L. Proposition 2.8. (a) (b = lim l)6(a = lim l) + (a; b); (b) d(b = lim l)6d(a = lim l) + (a; b). Deÿnition 2.10. The radius (the measure of divergence) r(l) of the sequence l is equal to r(l) = inf {(a = lim l); a ∈ L}: Lemma 2.10. r(l) = 0 if and only if l converges. Lemma 2.11. t = r(l) if only if for any k ∈ R+ there is a ∈ L such that almost all elements of l belong to the neighbourhood Op a = {c ∈ L; (a; c)¡p = t + k} of the point a. Lemma 2.11 states that r(l) is the radius of the least sphere in L, any spherical neighbourhood of which contains almost all elements from l.
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Deÿnition 2.11. A fuzzy subnumber M is a convex normalized fuzzy set M = (R; : R → I ) such that: (1) sup {(x); x ∈ R} = (a) for exactly one a ∈ R; (2) (x) is piecewise continuous. Deÿnition 2.12 (Zimmermann [11]). A fuzzy number M is a fuzzy subnumber for which sup {(x); x ∈ R} = 1. Proposition 2.9. If L is a complete metric space; then for any sequence l = {ai ; ai ∈ R; i ∈ !} the q-limit of l exists for q = (l). Proof. If such b ∈ L, which is equal to q-lim l, does not exist, then for any z¿q such d¿l exists that d = r-lim l. Let us consider a sequence h = {di ; i ∈ !} for which di = ri -lim l and (ri ; q)¡1=i. The sequence h either has limit points or does not have it. In the ÿrst case let d be a limit point of l. Then d = q-lim l. Really, for any k ∈ R+ there is dj ∈ h for which (dj ; d)¡ 19 k and dj = r-lim l with r −q¡1=k. Then for almost all ai ∈ l we have (ai ; d)6(ai ; dj ) + (dj ; d)¡r + 19 k¡q + 19 k + 19 k¡q + k. As k was chosen arbitrarily, d is a q-limit of l, and in this case the proposition is proved. Rn is a complete metric space. So, when the sequence h does not have limit points, there are such points 1 1 ; p−q¡ 10 . Then their neighbourhoods d; c ∈ h for which (d; c) = t¿2q+1; d = r-lim l; c = p-lim l; r −q¡ 10 1 1 Od = {a ∈ l; (a; d)¡r = 10 } and Oc = {a ∈ l; (a; c)¡p+ 10 } does not intersect, but according to Deÿnition 2.1 each of them has to contain almost all points of the sequence 1. This contradiction completes the proof of Proposition 2.7. Remark 2.5. Proposition 2.9 may be invalid if the supposition concerning L is not fulÿlled. It is demonstrated by the following example. Let L be the union of the two open intervals (−1; 0) and (1; 2) which contain the sequence l = {a2i = −1=i; a2i−1 = 1 + 1=i; i ∈ !}. For this sequence we have (l) = 1 but there are no (in L) 1-limit points of l. Proposition 2.10. If a = r-lim l and (a; b) = p; then b = (r + p)-lim l. Proposition 2.10 implies the following result. Proposition 2.11. (x = lim l) is a continuous function for an arbitrary sequence l. Proposition 2.9 and Deÿnition 2.11 imply the following result. Proposition 2.12. For any sequence l = {ai ; ai ∈ R; i ∈ !} the fuzzy set lim l is a fuzzy subnumber. Proposition 2.10 and Lemma 2.2 imply the following result. Theorem 2.6. For any sequence l = {ai ; ai ∈ R; i ∈ !} the fuzzy set lim l is a fuzzy number if and only if the sequence l converges. Let h = {bi ; bi ∈ L; i ∈ !} be a sequence in L and L be a normed real vector space [9].
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Theorem 2.7. If a = r-lim l and b = q-lim h then (a) a + b = (r + q)-lim (l + h) where l + h = {ai + bi ; i ∈ !}; (b) a − b = (r + q)-lim (l − h) where l − h = {ai − bi ; i ∈ !}; (c) ka = |k| · r-lim (kl) for any k ∈ R where kl = {kai ; i ∈ !}. Proof. (a) By the axiom of a triangle which is valid for any norm [9], we have (ai + bi ; a + b) = k(a + b) − (ai ; bi )k = k(a − ai ) + (b − bi )k6ka − ai k + kb − bi k = (ai ; a) + (b; bi ). So, (ai + bi ; a + b)6(ai ; a) + (b; bi ). As a consequence, (ai + bi ; a + b)6r + q for such indices i that (ai ; a)6r and (b; bi )6q. But these inequalities may be false only for a ÿnite set of indices. Thus, (ai + bi ; a + b)6r + q for almost all i ∈ !, and the ÿrst part of the theorem is proved. (b) The second part of the theorem is proved in a similar way. (c) For any k 6= 0 the conditions (ai ; a)6r and (kai ; ka)6|k| · r are equivalent in normed algebras [9]. Thus, ka = |k| · r-lim (kl) for any k ∈ R, and the theorem is proved. As 0 + 0 = 0 and k · 0 = 0 for any k ∈ R, we obtain Corollary 2.11. (Any course of the calculus, cf. [6, 7, 10]). If a = lim l; b = lim h then (a) a + b = lim (l + h); (b) a − b = lim (l − h); (c) ka = lim (kl). Corollary 2.12. If l; h are positive sequences, then (a + b = lim (l + h)) = (a = lim l) + (b = lim l). Corollary 2.13. If l; h are positive sequences, then (a − b = lim(l − h)) = |(a = lim l) − (b = lim l)|. Corollary 2.14. (ka = lim kl) = |k|(a = lim l). Corollary 2.15. If a is a fuzzy limit of l and b is a fuzzy limit of h; then (a + b) ((a − b), and ka) is a fuzzy limit of l + h (of (l − h), and kl), respectively. By the deÿnition (a + b = lim (l + h)) = =
1 1 = 1 + (a + b = lim (l + h)) 1 + (a = lim l) + (b = lim h) 1 1 − (a = lim l) 1 − (b = lim h) + 1+ (a = lim l) (b = lim h)
because (a = lim l) =
1 − (a = lim l) : (a = lim l)
So (a + b = lim (l + h)) =
(a = lim l) · (b = lim h) (a = lim l) · (b = lim h) + (b = lim h) − (a = lim l) · (b = lim h) + (a = lim l) − (a = lim l) · (b = lim h)
=
1 ; −1 (a = lim l) + −1 (b = lim h) − 1
and we obtain
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Corollary 2.16. (a + b = lim (l + h)) =
1 : −1 (a = lim l) + −1 (b = lim h) − 1
Corollary 2.17. (a − b = lim (l − h)) =
−1 (a = lim l)
1 : − −1 (b = lim h) − 1
Corollary 2.18. (ka = lim kl) =
(a = lim l) : |k| + (1 − |k|) · (a = lim l)
Let L be a normed algebra [9]. Theorem 2.8. If a = r-lim l and b = q-lim h; then ab = p-lim lh where lh = {ai ; bi ; i ∈ !} and p = min{(kak · q + sup kbi k · r); (kbk · r + sup kai k · q)}. 3. Local fuzzy limits of functions Let M; L be metric spaces with metrics which are in both cases denoted by the same letter ; f : M → L be an arbitrary function, and r ∈ R+ . Deÿnition 3.1. An element b ∈ L is called an r-limit of f at a point a ∈ M and denoted b = r-limx→ f(x) if for any sequence l satisfying the condition a = lim l, the equality b = r-lim f(l) is valid. Properties of fuzzy limits of number sequences make it possible to prove the following results. Proposition 3.1. An element b ∈ L is a limit of f(x) at a (in the classical sense [6, 10]) if and only if b = 0-limx→a f(x). Corollary 3.1. A 0-limit of a function is unique (if it exists). Proposition 3.2. If b = r-limx→a f(x); then b = q-limx→a f(x); for any q¿r. Let Lr (f; a) denote the set of all r-limits of the function f at the point a. Proposition 3.3. Lr (f; a) =
T a=lim l
Lr f(l).
The intersection of convex sets is a convex set [8]. Thus, Theorem 2.3 implies Theorem 3.1. Lr (f; a) is a convex set. Deÿnition 3.2. The upper defect (b = limx→a f(x)) of convergence of f(x) to b at a ∈ M is equal to o n inf r; b = lim f(x) : x→a
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Lemma 3.2. If q = (b = limx→a f(x)); then b = q-limx→a f(x). Proof. Let us consider some k ∈ R+ . By the deÿnition of an inÿmum, there is such r ∈ R that r − q¡t = 13 k and b = r-limx→a f(x). By the Deÿnition 3.1, there is such n ∈ ! that a = lim{ai } and (f(ai ); f(a))¡r + t for all i¿m. Thus, (f(ai ); f(a))¡r + t¡(q + t) + t6q + 2t¡q + k. As the number k was arbitrary, we have b = q-limx→a f(x). The lemma is proved. Deÿnition 3.3. The upper measure of convergence of f(x) to b at a ∈ M is equal to b = lim f(x) = x→a
1 : 1 + (b = limx→a f(x))
The measure of convergence of l deÿnes the fuzzy set limx→a f(x) = [L; (y = limx→a f(x))] of fuzzy limits of l. Deÿnition 3.4. (a) An element b ∈ L is called a fuzzy limit of a function f(x) at a point a ∈ L if (b = limx→a f(x))¡∞ or, equivalently, (b = limx→a f(x))¿0. (b) a function f(x) fuzzy converges at a point a ∈ L if it has a fuzzy limit at a point a ∈ L. Proposition 3.4. If a is a fuzzy limit of l and (a; b)¡∞; then b is a fuzzy limit of l. Deÿnition 3.5. An element a ∈ L is called a partial r-limit of f at a point a ∈ M (it is denoted a = rplimi→∞ ai ) if for any sequence l satisfying the condition a = lim l and any k ∈ R+ \{0}, the inequalities (b; f(ai ))6r + k are valid for inÿnitely many elements ai from l. Proposition 3.5. A function f has an r-limit at a point a if and only if f is bounded in some neighbourhood of a. Let L = M = E n . Theorem 3.2. If c = r-limx→a f(x) and d = q-limx→a g(x) then c ± d = (r + q)-limx→a (f ± g)(x). Remark. It is possible to think that c − d = (r − q)-limx→a (f − g)(x). However, it is not the case as the following example demonstrates: Let f(x) = x+1 = g(x); q = r = 1; c = 2; d = 0. Then c−d = 2 = (r+q)-limx→0 (f−g)(x) because f−g = 0, but c − d = 2 6= (r − q)-limx→0 (f − g)(x). Theorem 3.3. If r-limx→a f(x) = c and q ∈ R+ ; then rq-limx→a qf(x) = qc. 4. Conclusion We have demonstrated that in a broader context of fuzzy limits, it is possible to extend and make complete some basic results of the classical mathematical analysis. In the neoclassical analysis many results of classical mathematical analysis are reconsidered in a broader context of fuzzy set theory. This approach does not only bring new results which complete their classical analogues but also produces deeper insights and a better understanding of the classical theory.
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Acknowledgements The author expresses his gratitude to one of the referees for useful suggestions and comments. References [1] [2] [3] [4] [5] [6] [7] [8] [9] [10] [11]
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