aa r X i v : . [ m a t h . G M ] F e b Fuzzy Limits of Fuzzy Functions
Abdulhameed Qahtan Abbood Altai
University of babylon, Babil, Iraq 51002,[email protected]
February 7, 2019 abstract
In this paper, we propose the theory of fuzzy limit of fuzzy function depending on the Altai’sprinciple and using the representation theorem (resolution principle) to run the fuzzy arithmetic.
Keywords: fuzzy limit of fuzzy function, two-sided fuzzy limits, one-sided fuzzy limits, fuzzy limit atinfinite.
Zadeh introduced the concept of fuzzy set to assign to each object encountered in the real physical worldthat do not have precisely defined criteria of membership a grade of membership ranging between zero andone in 1965 [15]. Kramosil and Mich´ a lek defined the concept of fuzzy metric space using continuous t-normsin 1975 [9]. The fuzzy metric spaces have very important applications in quantum physics, particularly, inconnections with both string and ǫ ( ∞ ) theory which were studied by EI Naschie [14]. Matloka consideredbounded and convergent sequences of fuzzy numbers and studied their properties in 1986 [11]. Sequences offuzzy numbers also were discussed by Nanda [13], Kwon [10], Esi [5] and many others. Burgin introducedthe theory of fuzzy limits of functions based on the theory of fuzzy limits of sequences in 2000. He studiedand developed the construction of fuzzy limits of functions similar to the one of the fuzzy limits of sequencesbased on the concept of r − limit of function f [3]. In 2010, Altai defined the fuzzy metric spaces in a newway, that every real number r ∈ R is replaced by a fuzzy number r ∈ R , R = Z ∪ Q ∪ Q ′ , where if r ∈ Q ′ or r ∈ Q \ Z will be replaced by a triangular fuzzy number because of density of irrational and rational numbersin R and if r ∈ Z will be replaced by a singleton fuzzy set because of non density of integer numbers in R [1],and then using the representation theorem (resolution principle) to calculate the arithmetic operations acton α − cuts of fuzzy numbers [4]. And in 2011, Altai defined the limit fuzzy number of the convergent fuzzysequence in similar way [2]. Our goal is to establish the theory of fuzzy limits of fuzzy functions dependingon Altai’s principle, because it is very handy and convenient in the study of the fuzzy arithmetic. Representation theorem [4].
Let A be a fuzzy set in X with the membership function µ A ( x ). Let A α be the α − cuts of A and χ A α be the characteristic function of the crisp set A α , α ∈ (0 , µ A ( x ) = sup α ∈ (0 , ( α ∧ χ A α ( x )) , x ∈ X. Resolution principle [4].
Let A be a fuzzy set in X and αA α , α ∈ (0 ,
1] be a special fuzzy set, whosemembership function µ αA α ( x ) = ( α ∧ χ A α ( x )) , x ∈ X. Also, let Λ A = { α : µ A ( x ) = α for some x ∈ X }
1e the level set of A . Then A can be expressed in the form A = [ α ∈ Λ A ( αA α ) , where S denotes the standard fuzzy union. Remark [4].
The essence of representation theorem of fuzzy sets is that a fuzzy set A in X can be retrievedas a union of its αA α sets, α ∈ (0 ,
1] and the essence of resolution principle is that a fuzzy set A can bedecomposed into fuzzy sets αA α , α ∈ (0 ,
1] . Thus the representation theorem and the resolution principleare the same coin with two sides as both of them essentially tell that a fuzzy set A in X can always beexpressed in terms of its α − cuts without explicitly resorting to its membership function µ A ( x ). Proposition [1]
Let A be a fuzzy number, then A α is a closed, convex and compact subset of R , for all α ∈ (0 , In this section, definition of the fuzzy limit of fuzzy functions will be introduced and its properties will beconsidered.
Theorem 2.1.
Let (cid:16)
X, ρ (cid:17) and (cid:16)
Y , d (cid:17) be fuzzy metric spaces. Suppose that f : E ⊂ X → Y and p is afuzzy limit point of E . If for all α ∈ (0 , α − cut of f ( x ) converge to the bounds of α − cut of L , then f ( x ) converges to L ∈ Y as x → p . Proof.
For all α ∈ (0 , f ( x ,α , x ,α ) , f ( x ,α , x ,α )] , [ L ,α , L ,α ] be α − cuts of f ( x ) and L respectively,such that for all ε >
0, there exists δ , δ > < ρ (( x ,α , x ,α ) , ( p ,α , p ,α )) < δ ⇒ d ( f ( x ,α , x ,α ) , L i,α ) < ε, < ρ (( x ,α , x ,α ) , ( p ,α , p ,α )) < δ ⇒ d ( f ( x ,α , x ,α ) , L i,α ) < ε, where d ( f ( x ,α , x ,α ) , L i,α ) = min { d ( f ( x ,α , x ,α ) , L i,α ) : i = 1 , } ,d ( f ( x ,α , x ,α ) , L i,α ) = max { d ( f ( x ,α , x ,α ) , L i,α ) : i = 1 , } ,ρ (( x ,α , x ,α ) , ( p ,α , p ,α )) = min { ρ (( x ,α , x ,α ) , ( p ,α , p ,α )) : i = 1 , } ,ρ (( x ,α , x ,α ) , ( p ,α , p ,α )) = max { ρ (( x ,α , x ,α ) , ( p ,α , p ,α )) : i = 1 , } . If f ∗ ( x ,α , x ,α ) ∈ [ f ( x ,α , x ,α ) , f ( x ,α , x ,α )], by the squeeze theorem for functions that0 < ρ ∗ (( x ,α , x ,α ) , ( p ,α , p ,α )) < δ ∗ ⇒ d ∗ ( f ∗ ( x ,α , x ,α ) , L ∗ ,α ) < ε, where δ ∗ = min { δ , δ } and L ∗ ,α ∈ [ L ,α , L ,α ]. That is, the α − cut [ f ( x ,α , x ,α ) , f ( x ,α , x ,α )] of f ( x )converges to the α − cut [ L ,α , L ,α ] of L as the α − cut [ x ,α , x ,α ] of x approaches the α − cut [ p ,α , p ,α ] of p for all α ∈ (0 , (cid:3) Theorem 2.2.
Let (cid:16)
X, ρ (cid:17) and (cid:16)
Y , d (cid:17) be fuzzy metric spaces. Suppose that f : E ⊂ X → Y and p is afuzzy limit point of E . Then f ( x ) converges to L ∈ Y as x → p if and only if for all α ∈ (0 , ε > δ >
0, 0 < (cid:13)(cid:13)(cid:0) ρ (( x ,α , x ,α ) , ( p ,α , p ,α )) , ρ (( x ,α , x ,α ) , ( p ,α , p ,α )) (cid:1)(cid:13)(cid:13) < δ ⇒ k ( d ( f ( x ,α , x ,α ) , L i,α ) , d ( f ( x ,α , x ,α ) , L i,α )) k < ε, (2.1)2 roof. Let f ( x ) converge to L ∈ Y as x → p . By theorem 2.1, for all α ∈ (0 , ε >
0, there exists δ , δ >
0, 0 < ρ (( x ,α , x ,α ) , ( p ,α , p ,α )) < δ (cid:14) √ ⇒ d ( f ( x ,α , x ,α ) , L i,α ) < ε (cid:14) √ , < ρ (( x ,α , x ,α ) , ( p ,α , p ,α )) < δ (cid:14) √ ⇒ d ( f ( x ,α , x ,α ) , L i,α ) < ε (cid:14) √ . Then 0 < (cid:13)(cid:13)(cid:0) ρ (( x ,α , x ,α ) , ( p ,α , p ,α )) , ρ (( x ,α , x ,α ) , ( p ,α , p ,α )) (cid:1)(cid:13)(cid:13) = (cid:16) ( ρ (( x ,α , x ,α ) , ( p ,α , p ,α ))) + ( ρ (( x ,α , x ,α ) , ( p ,α , p ,α ))) (cid:17) / < δ where δ = min { δ , δ } , implies k ( d ( f ( x ,α , x ,α ) , L i,α ) , d ( f ( x ,α , x ,α ) , L i,α )) k = (cid:16) ( d ( f ( x ,α , x ,α ) , L i,α )) + ( d ( f ( x ,α , x ,α ) , L i,α )) (cid:17) / < ε. Now suppose (2.1) is given. Since ρ (( x ,α , x ,α ) , ( p ,α , p ,α )) ≤ (cid:13)(cid:13)(cid:0) ρ (( x ,α , x ,α ) , ( p ,α , p ,α )) , ρ (( x ,α , x ,α ) , ( p ,α , p ,α )) (cid:1)(cid:13)(cid:13) ; ρ (( x ,α , x ,α ) , ( p ,α , p ,α )) ≤ (cid:13)(cid:13)(cid:0) ρ (( x ,α , x ,α ) , ( p ,α , p ,α )) , ρ (( x ,α , x ,α ) , ( p ,α , p ,α )) (cid:1)(cid:13)(cid:13) and d ( f ( x ,α , x ,α ) , L i,α ) ≤ k ( d ( f ( x ,α , x ,α ) , L i,α ) , d ( f ( x ,α , x ,α ) , L i,α )) k ; d ( f ( x ,α , x ,α ) , L i,α ) ≤ k ( d ( f ( x ,α , x ,α ) , L i,α ) , d ( f ( x ,α , x ,α ) , L i,α )) k . Then 0 < ρ (( x ,α , x ,α ) , ( p ,α , p ,α )) < δ ⇒ d ( f ( x ,α , x ,α ) , L i,α ) < ε ;0 < ρ (( x ,α , x ,α ) , ( p ,α , p ,α )) < δ ⇒ d ( f ( x ,α , x ,α ) , L i,α ) < ε. (cid:3) Remark 2.1.
We will call L in theorem 2.2 by the fuzzy limit of f at p and write it as f ( p ) = L = lim x → p f ( x ) . (2.2) Examples 2.1.
1. To find the limit of f ( x ) = x − x +1 , as x → (0 , , α ∈ (0 , α − cut[ x ,α , x ,α ] − [4 , x ,α , x ,α ] + [1 ,
1] = (cid:20) min i,j,k =1 , (cid:26) x i,α x j,α x k,α − x i,α x j,α + 1 (cid:27) , max i,j,k =1 , (cid:26) x i,α x j,α x k,α − x i,α x j,α + 1 (cid:27)(cid:21) of f ( x )has the limit lim x ,α → αx ,α → − α min i,j,k =1 , (cid:26) x i,α x j,α x k,α − x i,α x j,α + 1 (cid:27) , lim x ,α → αx ,α → − α max i,j,k =1 , (cid:26) x i,α x j,α x k,α − x i,α x j,α + 1 (cid:27) . Taking the union of above α − cut we get the limit of the function.3. If f ( x ) = x + b, x ∈ R , then lim x → p f ( x ) = f ( p ) because, by the resolution principle, for all α ∈ (0 , ε >
0, there exists an δ >
0, 0 < k ( | x ,α − p ,α | , | x ,α − p ,α | ) k < δ ⇒ k ( | f ( x ,α , x ,α ) − f ( p ,α , p ,α ) | , | f ( x ,α , x ,α ) − f ( p ,α , p ,α ) | ) k = (cid:13)(cid:13)(cid:0) | ( x ,α + b ,α ) − ( p ,α + b ,α ) | , | ( x ,α − p ,α ) − ( p ,α + b ,α ) | (cid:1)(cid:13)(cid:13) ≤k ( | ( x ,α − p ,α ) | , | x ,α − p ,α | ) k + k ( | ( b ,α − b ,α ) | , | b ,α − b ,α | ) k < ( δ ,if b ∈ Z ,δ + k ( | ( b ,α − b ,α ) | , | b ,α − b ,α | ) k ,if b Z .
3. If f ( x ) = x + x − , x ∈ R , then lim x → f ( x ) = − α ∈ (0 , ε >
0, there exists 0 < δ ≤ < k ( | x ,α − | , | x ,α − | ) k < δ ⇒ k ( | f ( x ,α , x ,α ) − f (1 , | , | f ( x ,α , x ,α ) − f (1 , | ) k = k ( | y ,α + x ,α − | , | y ,α + x ,α − | ) k < √ δ = ε where y ,α = min { x ,α , x ,α x ,α , x ,α } ; y ,α = max { x ,α , x ,α x ,α , x ,α } and | y ,α + x ,α − | ≤ | x ,α − | | x ,α + 2 | < ( | x ,α | + 2) δ < δ, if y ,α = x ,α ; | y ,α + x ,α − | ≤ | x ,α − | | x ,α + 1 | + | x ,α − | < ( | x ,α | + 1) δ + δ < δ, if y ,α = x ,α x ,α ; | y ,α + x ,α − | ≤ (cid:12)(cid:12) x ,α − (cid:12)(cid:12) + | x ,α − | < ( | x ,α | + 1) δ + δ < δ, if y ,α = x ,α ; | y ,α + x ,α − | ≤ (cid:12)(cid:12) x ,α − (cid:12)(cid:12) + | x ,α − | < ( | x ,α | + 1) δ + δ < δ, if y ,α = x ,α ; | y ,α + x ,α − | ≤ | x ,α − | | x ,α + 1 | + | x ,α − | < ( | x ,α | + 1) δ + δ < δ, if y ,α = x ,α x ,α ; | y ,α + x ,α − | ≤ | x ,α − | | x ,α + 2 | < ( | x ,α | + 2) δ < δ, if y ,α = x ,α . Set δ = min (cid:8) , ε/ √ (cid:9) , we complete the proof.Now, we can consider basic properties of fuzzy limits of fuzzy functions and prove them depending on theabove theorems. Theorem 2.3.
The fuzzy limit of a fuzzy function is unique if it exists.
Proof.
Suppose f : E ⊂ X → Y and p ∈ X is a fuzzy limit point of E . Assume that lim x → p f ( x ) = L ; lim x → p f ( x ) = M .
So, by the resolution principle, for all α ∈ (0 , ε >
0, there exit δ , δ >
0, suchthat 0 < k ( ρ (( x ,α , x ,α ) , ( p ,α , p ,α )) , ρ (( x ,α , x ,α ) , ( p ,α , p ,α ))) k < δ ⇒ k ( d ( f ( x ,α , x ,α ) , L i,α ) , d ( f ( x ,α , x ,α ) , L i,α )) k < ε < k ( ρ (( x ,α , x ,α ) , ( p ,α , p ,α )) , ρ (( x ,α , x ,α ) , ( p ,α , p ,α ))) k < δ ⇒ k ( d ( f ( x ,α , x ,α ) , M i,α ) , d ( f ( x ,α , x ,α ) , M i,α )) k < ε . Let δ = min { δ , δ } . Then, for all α ∈ (0 , α − cut [ p ,α , p ,α ] of p satisfies0 < k ( ρ (( x ,α , x ,α ) , ( p ,α , p ,α )) , ρ (( x ,α , x ,α ) , ( p ,α , p ,α ))) k < δ ⇒ k ( d ( L i,α , M i,α ) , d ( L i,α , M i,α )) k ≤ k ( d ( L i,α , f ( x ,α , x ,α )) , d ( L i,α , f ( x ,α , x ,α ))) k + k ( d ( f ( x ,α , x ,α ) , M i,α ) , d ( f ( x ,α , x ,α ) , M i,α )) k < ε, where d ( L i,α , M i,α ) = min { d ( L i,α , M i,α ) : i = 1 , } , d ( L i,α , M i,α ) = max { d ( L i,α , M i,α ) : i = 1 , } . (cid:3) heorem 2.4. Let f : E ⊂ X → Y and p be a fuzzy limit point of E . Then lim x → p f ( x ) = L if and only iflim n →∞ f ( p n ) = L for every fuzzy sequence p n in E such that p n = p, lim n →∞ p n = p . Proof.
Suppose that lim x → p f ( x ) = L holds. By the resolution principle, for all α ∈ (0 , ε >
0, thereexists δ >
0, 0 < k ( ρ (( x ,α , x ,α ) , ( p ,α , p ,α )) , ρ (( x ,α , x ,α ) , ( p ,α , p ,α ))) k < δ ⇒ k ( d ( f ( x ,α , x ,α ) , L i,α ) , d ( f ( x ,α , x ,α ) , L i,α )) k < ε. Since p n → p , then for all α ∈ (0 , N ∈ N such that for n > N ,0 < k ( ρ (( p n, ,α , p n, ,α ) , ( p ,α , p ,α )) , ρ (( p n, ,α , p n, ,α ) , ( p ,α , p ,α ))) k < δ ⇒ k ( d ( f ( p n, ,α , p n, ,α ) , L i,α ) , d ( f ( p n, ,α , p n, ,α ) , L i,α )) k < ε. Conversely, assume lim n →∞ f ( p n ) = L but lim x → p f ( x ) = L . That is, there exists ε o >
0, such that for every δ >
0, that 0 < k ( ρ (( x ,α , x ,α ) , ( p ,α , p ,α )) , ρ (( x ,α , x ,α ) , ( p ,α , p ,α ))) k < δ but k ( d ( f ( x ,α , x ,α ) , L i,α ) , d ( f ( x ,α , x ,α ) , L i,α )) k > ε o Taking δ = n , n ∈ N , there is a p n in E such that0 < k ( ρ (( p n, ,α , p n, ,α ) , ( p ,α , p ,α )) , ρ (( p n, ,α , p n, ,α ) , ( p ,α , p ,α ))) k < n but k ( d ( f ( p n, ,α , p n, ,α ) , L i,α ) , d ( f ( p n, ,α , p n, ,α ) , L i,α )) k > ε o which contradicts the assumption lim n →∞ f ( p n ) = L . (cid:3) Theorem 2.5. If f and g are fuzzy functions such that lim x → p g ( x ) = L and lim u → L f ( u ) = f (cid:0) L (cid:1) , thenlim x → p f ( g ( x )) = f (cid:18) lim x → p g ( x ) (cid:19) = f (cid:0) L (cid:1) . Proof.
Since f ( u ) → f (cid:0) L (cid:1) as u → L , then by the resolution principle, for all α ∈ (0 , ε >
0, thereexits δ >
0, such that 0 < k ( ρ (( u ,α , u ,α ) , ( L ,α , L ,α )) , ρ (( u ,α , u ,α ) , ( L ,α , L ,α ))) k < δ ⇒ k ( d ( f ( u ,α , u ,α ) , f i ( L ,α , L ,α )) , d ( f ( u ,α , u ,α ) , f i ( L ,α , L ,α ))) k < ε. Since g ( x ) → L as x → p , then by the resolution principle, for all α ∈ (0 , δ ′ > < k ( σ (( x ,α , x ,α ) , ( p ,α , p ,α )) , σ (( x ,α , x ,α ) , ( p ,α , p ,α ))) k < δ ′ ⇒ k ( ρ ( g ( x ,α , x ,α ) , g i ( p ,α , p ,α )) , ρ ( g ( x ,α , x ,α ) , g i ( p ,α , p ,α ))) k < δ. Letting u ,α = g ( x ,α , x ,α ) , u ,α = g ( x ,α , x ,α ), we obtain0 < k ( σ (( x ,α , x ,α ) , ( p ,α , p ,α )) , σ (( x ,α , x ,α ) , ( p ,α , p ,α ))) k < δ ′ ⇒ (cid:13)(cid:13)(cid:13)(cid:0) d (cid:0) f ( g ( x ,α , x ,α ) , g ( x ,α , x ,α )) , f i ( L ,α , L ,α ) (cid:1) , d (cid:0) f ( g ( x ,α , x ,α ) , g ( x ,α , x ,α )) , f i ( L ,α , L ,α ) (cid:1)(cid:1)(cid:13)(cid:13) < ε. (cid:3) heorem 2.6. If E ⊂ R is a fuzzy metric space, p is a fuzzy limit point of E , f and g are fuzzy functionson E , and lim x → p f ( x ) and lim x → p g ( x ) are exist, then1. lim x → p ( f ( x ) + g ( x )) = lim x → p f ( x ) + lim x → p g ( x )2. lim x → p (cid:0) Af (cid:1) ( x ) = A lim x → p f ( x ) , A ∈ R
3. lim x → p ( f g )( x ) = lim x → p f ( x ) lim x → p g ( x )4. lim x → p (cid:16) f ( x ) g ( x ) (cid:17) = lim x → p f ( x )lim x → p g ( x ) . Proof.
For (1) and (2), by the resolution principle, we havelim x → p ( f ( x ) + g ( x )) = [ α ∈ (0 , α " lim x ,α → p ,α x ,α → p ,α ( f ( x ,α , x ,α ) + g ( x ,α , x ,α )) , lim x ,α → p ,α x ,α → p ,α ( f ( x ,α , x ,α ) + g ( x ,α , x ,α )) = [ α ∈ (0 , α " lim x ,α → p ,α x ,α → p ,α f ( x ,α , x ,α ) , lim x ,α → p ,α x ,α → p ,α f ( x ,α , x ,α ) + [ α ∈ (0 , α " lim x ,α → p ,α x ,α → p ,α g ( x ,α , x ,α ) , lim x ,α → p ,α x ,α → p ,α g ( x ,α , x ,α ) = lim x → p f ( x ) + lim x → p g ( x )and lim x → p (cid:0) Af (cid:1) ( x ) = [ α ∈ (0 , α " lim x ,α → p ,α x ,α → p ,α F ( x ,α , x ,α ) , lim x ,α → p ,α x ,α → p ,α F ( x ,α , x ,α ) = [ α ∈ (0 , ( α [ A ,α , A ,α ]) [ α ∈ (0 , α " lim x ,α → p ,α x ,α → p ,α f ( x ,α , x ,α ) , lim x ,α → p ,α x ,α → p ,α f ( x ,α , x ,α ) = A lim x → p f ( x )where F ( x ,α , x ,α ) = min { A ,α f ( x ,α , x ,α ) , A ,α f ( x ,α , x ,α ) , A ,α f ( x ,α , x ,α ) , A ,α f ( x ,α , x ,α ) } ,F ( x ,α , x ,α ) = max { A ,α f ( x ,α , x ,α ) , A ,α f ( x ,α , x ,α ) , A ,α f ( x ,α , x ,α ) , A ,α f ( x ,α , x ,α ) } . To prove (3), let lim x → p f ( x ) = L and lim x → p g ( x ) = M , then lim x → p (cid:2) f ( x ) − L (cid:3) = 0 and lim x → p (cid:2) g ( x ) − M (cid:3) = 0. Bythe resolution principle, for all α ∈ (0 , ε >
0, there exists δ >
0, such that0 < (cid:13)(cid:13)(cid:0) | x ,α − p ,α | , | x ,α − p ,α | (cid:1)(cid:13)(cid:13) < δ ⇒ k ( | f ( x ,α , x ,α ) − L ,α | , | f ( x ,α , x ,α ) − L ,α | ) k < ε ;0 < (cid:13)(cid:13)(cid:0) | x ,α − p ,α | , | x ,α − p ,α | (cid:1)(cid:13)(cid:13) < δ ⇒ k ( | g ( x ,α , x ,α ) − M ,α | , | g ( x ,α , x ,α ) − M ,α | ) k < ε. So, k ( | ( F G ) | , | ( F G ) | ) k ≤ k ( | F | , | F | ) k k ( | G | , | G | ) k < ε. F = f ( x ,α , x ,α ) − L ,α , F = f ( x ,α , x ,α ) − L ,α ,G = g ( x ,α , x ,α ) − M ,α , G = g ( x ,α , x ,α ) − M ,α , ( F G ) = min { F G , F G , F G , F G } , ( F G ) = max { F G , F G , F G , F G } . That is, lim x ,α → p ,α x ,α → p ,α ( F G ) = 0 , lim x ,α → p ,α x ,α → p ,α ( F G ) = 0 . From properties (1) and (2), if f ( x ,α , x ,α ) g ( x ,α , x ,α ) = min { f i ( x ,α , x ,α ) g i ( x , x ,α ) : i = 1 , } or f ( x ,α , x ,α ) g ( x ,α , x ,α ) = max { f i ( x ,α , x ,α ) g i ( x , x ,α ) : i = 1 , } , thenlim x ,α → p ,α x ,α → p ,α f ( x ,α , x ,α ) g ( x ,α , x ,α ) = lim x ,α → p ,α x ,α → p ,α (cid:16) [ f ( x ,α , x ,α ) − L ,α ] [ g ( x ,α , x ,α ) − M ,α ]+ L ,α g ( x ,α , x ,α ) + M ,α f ( x ,α , x ,α ) − L ,α M ,α (cid:17) = 0 + L ,α M ,α + L ,α M ,α − L ,α M ,α = L ,α M ,α . If f ( x ,α , x ,α ) g ( x ,α , x ,α ) = min { f i ( x ,α , x ,α ) g i ( x , x ,α ) : i = 1 , } or f ( x ,α , x ,α ) g ( x ,α , x ,α ) =max { f i ( x ,α , x ,α ) g i ( x , x ,α ) : i = 1 , } , thenlim x ,α → p ,α x ,α → p ,α f ( x ,α , x ,α ) g ( x ,α , x ,α ) = lim x ,α → p ,α x ,α → p ,α (cid:16) [ f ( x ,α , x ,α ) − L ,α ] [ g ( x ,α , x ,α ) − M ,α ]+ L ,α g ( x ,α , x ,α ) + M ,α f ( x ,α , x ,α ) − L ,α M ,α (cid:17) = 0 + L ,α M ,α + L ,α M ,α − L ,α M ,α = L ,α M ,α . If f ( x ,α , x ,α ) g ( x ,α , x ,α ) = min { f i ( x ,α , x ,α ) g i ( x , x ,α ) : i = 1 , } or f ( x ,α , x ,α ) g ( x ,α , x ,α ) =max { f i ( x ,α , x ,α ) g i ( x , x ,α ) : i = 1 , } , thenlim x ,α → p ,α x ,α → p ,α f ( x ,α , x ,α ) g ( x ,α , x ,α ) = lim x ,α → p ,α x ,α → p ,α (cid:16) [ f ( x ,α , x ,α ) − L ,α ] [ g ( x ,α , x ,α ) − M ,α ]+ L ,α g ( x ,α , x ,α ) + M ,α f ( x ,α , x ,α ) − L ,α M ,α (cid:17) = 0 + L ,α M ,α + L ,α M ,α − L ,α M ,α = L ,α M ,α . If f ( x ,α , x ,α ) g ( x ,α , x ,α ) = min { f i ( x ,α , x ,α ) g i ( x , x ,α ) : i = 1 , } or f ( x ,α , x ,α ) g ( x ,α , x ,α ) =max { f i ( x ,α , x ,α ) g i ( x , x ,α ) : i = 1 , } , thenlim x ,α → p ,α x ,α → p ,α f ( x ,α , x ,α ) g ( x ,α , x ,α ) = lim x ,α → p ,α x ,α → p ,α (cid:16) [ f ( x ,α , x ,α ) − L ,α ] [ g ( x ,α , x ,α ) − M ,α ]+ L ,α g ( x ,α , x ,α ) + M ,α f ( x ,α , x ,α ) − L ,α M ,α (cid:17) = 0 + L ,α M ,α + L ,α M ,α − L ,α M ,α = L ,α M ,α . Finally, since lim x → p g ( x ) = M , then by the resolution principle, for all α ∈ (0 , ε > δ > < k ( | x ,α − p ,α | , | x ,α − p ,α | ) k < δ ⇒ k ( | g ( x ,α , x ,α ) − M ,α | , | g ( x ,α , x ,α ) − M ,α | ) k < ε. So,0 < k ( | x ,α − p ,α | , | x ,α − p ,α | ) k < δ ⇒ k ( | g ( x ,α , x ,α ) − M ,α | , | g ( x ,α , x ,α ) − M ,α | ) k < k ( | M ,α | , | M ,α | ) k k ( | M ,α | , | M ,α | ) k ≤ k ( | g ( x ,α , x ,α ) | , | g ( x ,α , x ,α ) | ) k + k ( | g ( x ,α , x ,α ) − M ,α | , | g ( x ,α , x ,α ) − M ,α | ) k < k ( | g ( x ,α , x ,α ) − M ,α | , | g ( x ,α , x ,α ) − M ,α | ) k + k ( | M ,α | , | M ,α | ) k k ( | g ( x ,α , x ,α ) | , | g ( x ,α , x ,α ) | ) k < k ( | M ,α | , | M ,α | ) k . Also, there exists δ > < k ( | x ,α − p ,α | , | x ,α − p ,α | ) k < δ ⇒ k ( | g ( x ,α , x ,α ) − M ,α | , | g ( x ,α , x ,α ) − M ,α | ) k < k ( | M ,α | , | M ,α | ) k ε . Set δ = min { δ , δ } , then0 < k ( | x ,α − p ,α | , | x ,α − p ,α | ) k < δ ⇒ (cid:12)(cid:12)(cid:12)(cid:12) k ( | g ( x ,α , x ,α ) | , | g ( x ,α , x ,α ) | ) k − k ( | M ,α | , | M ,α | ) k (cid:12)(cid:12)(cid:12)(cid:12) = |k ( | M ,α | , | M ,α | ) k − k ( | g ( x ,α , x ,α ) | , | g ( x ,α , x ,α ) | ) k|k ( | g ( x ,α , x ,α ) | , | g ( x ,α , x ,α ) | ) k k ( | M ,α | , | M ,α | ) k ≤ k ( | g ( x ,α , x ,α ) − M ,α | , | g ( x ,α , x ,α ) − M ,α | ) kk ( | g ( x ,α , x ,α ) | , | g ( x ,α , x ,α ) | ) k k ( | M ,α | , | M ,α | ) k < k ( | M ,α | , | M ,α | ) k k ( | M ,α | , | M ,α | ) k ε ε. (cid:3) Theorem 2.7.
Let a ∈ I ⊂ R , where I is an open fuzzy interval. If f, g are fuzzy functions defined on I \ a such that f ( x ) = g ( x ) , x ∈ I \ a and f ( x ) → L as x → a , then lim x → a g ( x ) = lim x → a f ( x ) Proof.
Since f ( x ) → L as x → a , then by the resolution principle, for all α ∈ (0 , ε >
0, thereexists δ > < k ( | x ,α − p ,α | , | x ,α − p ,α | ) k < δ ⇒ k ( | f ( x ,α , x ,α ) − L ,α | , | f ( x ,α , x ,α ) − L ,α | ) k < ε ;Since f ( x ) = g ( x ) , x ∈ I \ a , then for all α ∈ (0 ,
1] that [ f ( x ,α , x ,α ) , f ( x ,α , x ,α )] = [ g ( x ,α , x ,α ) , g ( x ,α , x ,α )].Thus,0 < k ( | x ,α − p ,α | , | x ,α − p ,α | ) k < δ ⇒ k ( | g ( x ,α , x ,α ) − L ,α | , | g ( x ,α , x ,α ) − L ,α | ) k < ε. (cid:3) Theorem 2.8. Comparison theorem for fuzzy functions.
Suppose a ∈ I ⊂ R , where I is an openfuzzy interval, and f, g are fuzzy functions defined on I \ a . If f and g have limits as x → a and f ( x ) ≤ g ( x )for all x ∈ I \ a , then lim x → p f ( x ) ≤ lim x → p g ( x ). Proof.
Let lim x → p f ( x ) = L and lim x → p g ( x ) = M , and suppose that L > M . By the resolution principle, forall α ∈ (0 , L ,α , L ,α ] > [ M ,α , M ,α ]. Let ε > , ε > , ε + ε = [ L ,α − M ,α ], there exist δ > , δ > < k ( | x ,α − p ,α | , | x ,α − p ,α | ) k < δ ⇒ k ( | f ( x ,α , x ,α ) − L ,α | , | f ( x ,α , x ,α ) − L ,α | ) k < ε ;0 < k ( | x ,α − p ,α | , | x ,α − p ,α | ) k < δ ⇒ k ( | g ( x ,α , x ,α ) − M ,α | , | g ( x ,α , x ,α ) − M ,α | ) k < ε . δ = min { δ , δ } , we get0 < k ( | x ,α − p ,α | , | x ,α − p ,α | ) k < δ ⇒ ( f ( x ,α , x ,α ) − g ( x ,α , x ,α ) , f ( x ,α , x ,α ) − g ( x ,α , x ,α )) =( f ( x ,α , x ,α ) − L ,α , f ( x ,α , x ,α ) − L ,α ) + ( L ,α − M ,α , L ,α − M ,α )+( M ,α − g ( x ,α , x ,α ) , M ,α − g ( x ,α , x ,α )) > ( L ,α − M ,α − ε − ε , L ,α − M ,α − ε − ε ) > (0 , f ( x ) ≤ g ( x ) for all x ∈ I \ p . (cid:3) Theorem 2.9. Squeeze theorem for fuzzy functions.
Suppose p ∈ I ⊂ R , where I is an openfuzzy interval, and f, g, h are fuzzy functions defined on I \ p . If f ( x ) ≤ h ( x ) ≤ g ( x ) for all x ∈ I \ p , and lim x → p f ( x ) = lim x → p g ( x ) = L then lim x → p h ( x ) = L . Proof.
Since lim x → p f ( x ) = lim x → p g ( x ) = L , by the resolution principle, for all α ∈ (0 , ε >
0, thereexist δ > δ > < k ( | x ,α − p ,α | , | x ,α − p ,α | ) k < δ ⇒ k ( | f ( x ,α , x ,α ) − L ,α | , | f ( x ,α , x ,α ) − L ,α | ) k < ε ;0 < k ( | x ,α − p ,α | , | x ,α − p ,α | ) k < δ ⇒ k ( | g ( x ,α , x ,α ) − L ,α | , | g ( x ,α , x ,α ) − L ,α | ) k < ε. Since f ( x ) ≤ h ( x ) ≤ g ( x ) for all x ∈ I \ a , then by resolution principle, for all α ∈ (0 , δ > < k ( | x ,α − p ,α | , | x ,α − p ,α | ) k < δ ′′ ⇒ (cid:16) L ,α − ε (cid:14) √ , L ,α − ε (cid:14) √ (cid:17) < (cid:16) f ( x ,α , x ,α ) , f ( x ,α , x ,α ) (cid:17) ≤ (cid:16) h ( x ,α , x ,α ) , h ( x ,α , x ,α ) (cid:17) ≤ (cid:16) g ( x ,α , x ,α ) , g ( x ,α , x ,α ) (cid:17) < (cid:16) L ,α + ε (cid:14) √ , L ,α + ε (cid:14) √ (cid:17) . Choosing δ = min { δ , δ , δ } we have0 < (cid:13)(cid:13)(cid:0) | x ,α − p ,α | , | x ,α − p ,α | (cid:1)(cid:13)(cid:13) < δ ⇒ ( | h ( x ,α , x ,α ) − L ,α | , | h ( x ,α , x ,α ) − L ,α | ) < (cid:18) ε √ , ε √ (cid:19) which completes the proof. (cid:3) We try in this section to establish the concept of the one-side fuzzy limit of fuzzy functions through thefollowing theorem whose proofs are similar to proofs of theorems 2.1 and 2.2 respectively.
Theorem 3.1.
Let f : I ⊂ R → R be a fuzzy function defined on some open fuzzy interval I with leftendpoint p . Then f ( x ) converges to L as x approaches p from the right if for all α ∈ (0 , α − cut of f ( x ) converge to the bounds of α − cut of L as the bounds of α − cut of x approach from the rightto the bounds of α − cut of p . Theorem 3.2.
Let f : I ⊂ R → R be a fuzzy function defined on some open fuzzy interval I with rightendpoint p . Then f ( x ) converges to L as x approaches p from the left if for all α ∈ (0 , α − cut of f ( x ) converge to the bounds of α − cut of L as the bounds of α − cut of x approach from the left tothe bounds of α − cut of p . Theorem 3.3.
Let f : I ⊂ R → R be a fuzzy function defined on some open fuzzy interval I with leftendpoint p . Then f ( x ) converges to L as x approaches p from the right if and only if for all α ∈ (0 , ε >
0, there exists δ , δ > , < ( x ,α − p ,α , x ,α − p ,α ) < ( δ , δ ) ⇒ k ( | f ( x ,α , x ,α ) − L ,α | , | f ( x ,α , x ,α ) − L ,α | ) k < ε. (3.1)9 heorem 3.4. Let f : I ⊂ R → R be a fuzzy function defined on some open fuzzy interval I with rightendpoint p . Then f ( x ) converges to L as x approaches p from the left if and only if for all α ∈ (0 , ε >
0, there exists δ , δ > − δ , − δ ) < ( x ,α − p ,α , x ,α − p ,α ) < (0 , ⇒ k ( | f ( x ,α , x ,α ) − L ,α | , | f ( x ,α , x ,α ) − L ,α | ) k < ε. (3.2) Remark 3.1.
1. We will call L in theorem 3.3 by the right-hand fuzzy limit of f at p and write it as f ( p + ) = L = lim x → p + f ( x ) (3.3)if by the resolution principle, for all α ∈ (0 , ε >
0, there exists δ > L in theorem 3.4 by the left-hand fuzzy limit of f at p and write it as f ( p − ) = L = lim x → p − f ( x ) (3.4)if by the resolution principle, for all α ∈ (0 , ε >
0, there exists δ >
Examples 3.1.
1. Both lim x → ( , , ) + x − ( , , ) and lim x → ( , , ) − x − ( , , ) do not exist, because by the resolution principle,for all α ∈ (0 , ε > ε ′ >
0, there exist an δ , δ > δ ′ , δ ′ > , < (cid:18) x ,α − (cid:18) − α + 12 (cid:19) , x ,α − (cid:18) α + 14 (cid:19)(cid:19) < ( δ , δ ) ⇒ (cid:13)(cid:13)(cid:13)(cid:13)(cid:13) (cid:12)(cid:12)(cid:12)(cid:12)(cid:12) x ,α − (cid:0) α + (cid:1) − ∞ (cid:12)(cid:12)(cid:12)(cid:12)(cid:12) , (cid:12)(cid:12)(cid:12)(cid:12)(cid:12) x ,α − (cid:0) − α + (cid:1) − ∞ (cid:12)(cid:12)(cid:12)(cid:12)(cid:12)!(cid:13)(cid:13)(cid:13)(cid:13)(cid:13) > k (1 /δ , /δ ) k > ε ;( − δ ′ , − δ ′ ) < (cid:18) x ,α − (cid:18) − α + 12 (cid:19) , x ,α − (cid:18) α + 14 (cid:19)(cid:19) < (0 , ⇒ (cid:13)(cid:13)(cid:13)(cid:13)(cid:13) (cid:12)(cid:12)(cid:12)(cid:12)(cid:12) x ,α − (cid:0) α + (cid:1) − ∞ (cid:12)(cid:12)(cid:12)(cid:12)(cid:12) , (cid:12)(cid:12)(cid:12)(cid:12)(cid:12) x ,α − (cid:0) − α + (cid:1) − ∞ (cid:12)(cid:12)(cid:12)(cid:12)(cid:12)!(cid:13)(cid:13)(cid:13)(cid:13)(cid:13) > k (1 /δ ′ , /δ ′ ) k > ε ′ .
2. lim x → + exp (cid:0) /x (cid:1) does not exist but lim x → − exp (cid:0) /x (cid:1) exists, because by the resolution principle, for all α ∈ (0 , ε >
0, there exist an δ , δ > , < ( x ,α , x ,α ) < ( δ , δ ) ⇒k ( | exp (1 /x ,α ) − ∞| , | exp (1 /x ,α ) − ∞| ) k > k ( | exp (1 /δ ) − ∞| , | exp (1 /δ ) − ∞| ) k > ε, and for all ε ′ >
0, there exists δ ′ , δ ′ > − δ ′ , − δ ′ ) < ( x ,α , x ,α ) < (0 , ⇒ k ( | exp (1 /x ,α ) | , | exp (1 /x ,α ) | ) k < k ( | exp ( − /δ ′ ) | , | exp ( − /δ ′ ) | ) k < ε ′ .
3. The function f ( x ) = (cid:26) x , x < ( , , )( , , ) ,( , , ) < x has both lim x → ( , , ) − f ( x ) and lim x → ( , , ) + f ( x )because by the resolution principle, for all α ∈ (0 , ε >
0, there exists δ , δ > , < (cid:18) x ,α − (cid:18) − α + 14 (cid:19) , x ,α − (cid:18) α + 16 (cid:19)(cid:19) < ( δ , δ ) ⇒ (cid:13)(cid:13)(cid:13)(cid:13)(cid:12)(cid:12)(cid:12)(cid:12)(cid:18) α + 136 (cid:19) − (cid:18) α + 136 (cid:19)(cid:12)(cid:12)(cid:12)(cid:12) , (cid:12)(cid:12)(cid:12)(cid:12)(cid:18) − α + 116 (cid:19) − (cid:18) − α + 116 (cid:19)(cid:12)(cid:12)(cid:12)(cid:12)(cid:13)(cid:13)(cid:13)(cid:13) < ε α ∈ (0 , δ ′ , δ ′ > − δ ′ , − δ ′ ) < (cid:18) x ,α − (cid:18) − α + 14 (cid:19) , x ,α − (cid:18) α + 16 (cid:19)(cid:19) < (0 , (cid:12)(cid:12)(cid:12)(cid:12)(cid:12) x ,α − (cid:18) − α + 14 (cid:19) (cid:12)(cid:12)(cid:12)(cid:12)(cid:12) ≤ (cid:12)(cid:12)(cid:12)(cid:12) x ,α − (cid:18) − α + 14 (cid:19)(cid:12)(cid:12)(cid:12)(cid:12) (cid:12)(cid:12)(cid:12)(cid:12) x ,α + (cid:18) − α + 14 (cid:19)(cid:12)(cid:12)(cid:12)(cid:12) < (cid:18) | x ,α | + 2 (cid:18) − α + 14 (cid:19)(cid:19) δ < (cid:18) − α + 14 (cid:19) δ , (cid:12)(cid:12)(cid:12)(cid:12)(cid:12) x ,α − (cid:18) α + 16 (cid:19) (cid:12)(cid:12)(cid:12)(cid:12)(cid:12) ≤ (cid:12)(cid:12)(cid:12)(cid:12) x ,α − (cid:18) − α + 14 (cid:19)(cid:12)(cid:12)(cid:12)(cid:12) (cid:12)(cid:12)(cid:12)(cid:12) x ,α + (cid:18) − α + 14 (cid:19)(cid:12)(cid:12)(cid:12)(cid:12) + (cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:18) − α + 14 (cid:19) − (cid:18) α + 16 (cid:19) (cid:12)(cid:12)(cid:12)(cid:12)(cid:12) < (cid:18) − α + 14 (cid:19) δ + (cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:18) − α + 14 (cid:19) − (cid:18) α + 16 (cid:19) (cid:12)(cid:12)(cid:12)(cid:12)(cid:12) < (cid:18) − α + 14 (cid:19) δ , (cid:12)(cid:12)(cid:12)(cid:12)(cid:12) x ,α − (cid:18) α + 16 (cid:19) (cid:12)(cid:12)(cid:12)(cid:12)(cid:12) ≤ (cid:12)(cid:12)(cid:12)(cid:12) x ,α − (cid:18) α + 16 (cid:19)(cid:12)(cid:12)(cid:12)(cid:12) (cid:12)(cid:12)(cid:12)(cid:12) x ,α + (cid:18) α + 16 (cid:19)(cid:12)(cid:12)(cid:12)(cid:12) < (cid:18) | x ,α | + 2 (cid:18) α + 16 (cid:19)(cid:19) δ < (cid:18) α + 16 (cid:19) δ , (cid:12)(cid:12)(cid:12)(cid:12)(cid:12) x ,α − (cid:18) − α + 14 (cid:19) (cid:12)(cid:12)(cid:12)(cid:12)(cid:12) ≤ (cid:12)(cid:12)(cid:12)(cid:12) x ,α − (cid:18) α + 16 (cid:19)(cid:12)(cid:12)(cid:12)(cid:12) (cid:12)(cid:12)(cid:12)(cid:12) x ,α + (cid:18) α + 16 (cid:19)(cid:12)(cid:12)(cid:12)(cid:12) + (cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:18) α + 16 (cid:19) − (cid:18) − α + 14 (cid:19) (cid:12)(cid:12)(cid:12)(cid:12)(cid:12) < (cid:18) α + 16 (cid:19) δ + (cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:18) α + 16 (cid:19) − (cid:18) − α + 14 (cid:19) (cid:12)(cid:12)(cid:12)(cid:12)(cid:12) < (cid:18) α + 16 (cid:19) δ , (cid:12)(cid:12)(cid:12)(cid:12)(cid:12) x ,α x ,α − (cid:18) − α + 14 (cid:19) (cid:12)(cid:12)(cid:12)(cid:12)(cid:12) ≤ (cid:20)(cid:12)(cid:12)(cid:12)(cid:12) x ,α − (cid:18) − α + 14 (cid:19)(cid:12)(cid:12)(cid:12)(cid:12) + (cid:18) − α + 14 (cid:19)(cid:21) (cid:20)(cid:12)(cid:12)(cid:12)(cid:12) x ,α − (cid:18) α + 16 (cid:19)(cid:12)(cid:12)(cid:12)(cid:12) + (cid:18) α + 16 (cid:19)(cid:21) + (cid:18) − α + 14 (cid:19) < (cid:20) δ + (cid:18) − α + 14 (cid:19)(cid:21) (cid:20) δ + (cid:18) α + 16 (cid:19)(cid:21) + (cid:18) − α + 14 (cid:19) , (cid:12)(cid:12)(cid:12)(cid:12)(cid:12) x ,α x ,α − (cid:18) α + 16 (cid:19) (cid:12)(cid:12)(cid:12)(cid:12)(cid:12) ≤ (cid:20)(cid:12)(cid:12)(cid:12)(cid:12) x ,α − (cid:18) − α + 14 (cid:19)(cid:12)(cid:12)(cid:12)(cid:12) + (cid:18) − α + 14 (cid:19)(cid:21) (cid:20)(cid:12)(cid:12)(cid:12)(cid:12) x ,α − (cid:18) α + 16 (cid:19)(cid:12)(cid:12)(cid:12)(cid:12) + (cid:18) α + 16 (cid:19)(cid:21) + (cid:18) α + 16 (cid:19) < (cid:20) δ + (cid:18) − α + 14 (cid:19)(cid:21) (cid:20) δ + (cid:18) α + 16 (cid:19)(cid:21) + (cid:18) α + 16 (cid:19) . then, by considering above various cases, for all ε ′ ( δ , δ ) >
0, we get (cid:13)(cid:13)(cid:13)(cid:13)(cid:13)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12) y ,α − (cid:18) − α + 14 (cid:19) (cid:12)(cid:12)(cid:12)(cid:12)(cid:12) , (cid:12)(cid:12)(cid:12)(cid:12)(cid:12) y ,α − (cid:18) α + 16 (cid:19) (cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:13)(cid:13)(cid:13)(cid:13)(cid:13) < ε ′ , where y ,α = min { x ,α , x ,α x ,α , x ,α } , y ,α = max { x ,α , x ,α x ,α , x ,α } and Theorem 3.5. f : R → R be a fuzzy function, then lim x → p f ( x ) = L if and only if L = lim x → p − f ( x ) = lim x → p + f ( x ) . roof. Suppose that lim x → p f ( x ) = L . By the resolution principle, for all α ∈ (0 , ε >
0, there exists δ > < k ( | x ,α − p ,α | , | x ,α − p ,α | ) k < δ ⇒ k ( | f ( x ,α , x ,α ) − L ,α | , | f ( x ,α , x ,α ) − L ,α | ) k < ε. Since for all α ∈ (0 , , < ( x ,α − p ,α , x ,α − p ,α ) < ( δ , δ ) and ( − δ , − δ ) < ( x ,α − p ,α , x ,α − p ,α ) < (0 , < k ( | x ,α − p ,α | , | x ,α − p ,α | ) k < δ, then(0 , < ( x ,α − p ,α , x ,α − p ,α ) < ( δ / √ , δ / √ ⇒ k ( | f ( x ,α , x ,α ) − L ,α | , | f ( x ,α , x ,α ) − L ,α | ) k < ε ;( − δ / √ , − δ / √ < ( x ,α − p ,α , x ,α − p ,α ) < (0 , ⇒ k ( | f ( x ,α , x ,α ) − L ,α | , | f ( x ,α , x ,α ) − L ,α | ) k < ε. Conversely, suppose L = lim x → p − f ( x ) = lim x → p + f ( x ) holds. By the resolution principle, for all α ∈ (0 , ε >
0, there exists δ > δ ′ > , < ( x ,α − p ,α , x ,α − p ,α ) < ( δ / √ , δ / √ ⇒ k ( | f ( x ,α , x ,α ) − L ,α | , | f ( x ,α , x ,α ) − L ,α | ) k < ε, ( − δ / √ , − δ / √ < ( x ,α − p ,α , x ,α − p ,α ) < (0 , ⇒ k ( | f ( x ,α , x ,α ) − L ,α | , | f ( x ,α , x ,α ) − L ,α | ) k < ε Set δ = min { δ , δ } . Then0 < (cid:13)(cid:13)(cid:0) | x ,α − p ,α | , | x ,α − p ,α | (cid:1)(cid:13)(cid:13) < δ ⇒ k ( | f ( x ,α , x ,α ) − L ,α | , | f ( x ,α , x ,α ) − L ,α | ) k < ε. (cid:3) Examples 3.2.
1. The function f ( x ) = | sin( x ) | sin( x ) has no fuzzy limit at 0 because by the resolution principle, for all α ∈ (0 , α − cuts lim x ,α → + x ,α → + min n | sin( x i,α ) | sin( x i,α ) : i = 1 , o and lim x ,α → + x ,α → + max n | sin( x i,α ) | sin( x i,α ) : i = 1 , o give pos-itive values and lim x ,α → − x ,α → − min n | sin( x i,α ) | sin( x i,α ) : i = 1 , o and lim x ,α → − x ,α → − max n | sin( x i,α ) | sin( x i,α ) : i = 1 , o give negativevalues.2. The function f ( x ) = x + 1 , x >
15 , x = 17 x − x < x = 1 because by the resolutionprinciple, for all α ∈ (0 , α − cuts lim x ,α → + x ,α → + (2 x ,α + 1) , lim x ,α → + x ,α → + (2 x ,α + 1) = [3 ,
3] ; lim x ,α → − x ,α → − (7 y ,α − , lim x ,α → − x ,α → − (7 y ,α − = [3 , . where y ,α = min { x i,α x j,α : i, j = 1 , } ; y ,α = max { x i,α x j,α : i, j = 1 , } . Thus, lim x → + f ( x ) =3; lim x → − f ( x ) = 3, and by theorem 3.5, lim x → f ( x ) = 3.12 Fuzzy limit at infinity. x → ±∞ Concept of fuzzy limit of fuzzy function at infinity will be given here through the following theorems whoseproofs are similar to proofs of theorems 2.1 and 2.2.
Theorem 4.1.1.
Let f : E ⊂ R → R be a fuzzy function and ( a, ∞ ) ⊆ E for some a ∈ R . Then f ( x )converges to L ∈ R as x approaches ∞ if for all α ∈ (0 , α − cut of f ( x ) converge to thebounds of α − cut of L as the bounds of α − cut of x approach ∞ . Theorem 4.1.2.
Let f : E ⊂ R → R be a fuzzy function and ( −∞ , a ) ⊆ E for some a ∈ R . Then f ( x )converges to L ∈ R as x approaches −∞ if for all α ∈ (0 , α − cut of f ( x ) converge to thebounds of α − cut of L as the bounds of α − cut of x approach −∞ . Theorem 4.1.3.
Let f : E ⊂ R → R be a fuzzy function and ( a, ∞ ) ⊆ E for some a ∈ R . Then f ( x )converges to L as x approaches ∞ if and only if for all α ∈ (0 , ε >
0, there exists K such that the α − cuts [ K ,α , K ,α ] of K , [ x ,α , x ,α ] of x and (cid:2)(cid:12)(cid:12) f ( x ,α , x ,α ) − L ,α (cid:12)(cid:12) , (cid:12)(cid:12) f ( x ,α , x ,α ) − L ,α (cid:12)(cid:12)(cid:3) of (cid:12)(cid:12) f ( x ) − L (cid:12)(cid:12) satisfy that K ,α = K ,α ( ε ) > a ,α , K ,α = K ,α ( ε ) > a ,α and( x ,α , x ,α ) > ( K ,α , K ,α ) ⇒ (cid:13)(cid:13)(cid:0)(cid:12)(cid:12) f ( x ,α , x ,α ) − L ,α (cid:12)(cid:12) , (cid:12)(cid:12) f ( x ,α , x ,α ) − L ,α (cid:12)(cid:12)(cid:1)(cid:13)(cid:13) < ε. (4.1) Theorem 4.1.4.
Let f : E ⊂ R → R be a fuzzy function and ( a, ∞ ) ⊆ E for some a ∈ R . Then f ( x )converges to L as x approaches ∞ if and only if for all α ∈ (0 , ε >
0, there exists K such that the α − cuts [ K ,α , K ,α ] of K , [ x ,α , x ,α ] of x and (cid:2)(cid:12)(cid:12) f ( x ,α , x ,α ) − L ,α (cid:12)(cid:12) , (cid:12)(cid:12) f ( x ,α , x ,α ) − L ,α (cid:12)(cid:12)(cid:3) of (cid:12)(cid:12) f ( x ) − L (cid:12)(cid:12) satisfy that K ,α = K ,α ( ε ) < a ,α , K ,α = K ,α ( ε ) < a ,α and( x ,α , x ,α ) < ( K ,α , K ,α ) ⇒ (cid:13)(cid:13)(cid:0)(cid:12)(cid:12) f ( x ,α , x ,α ) − L ,α (cid:12)(cid:12) , (cid:12)(cid:12) f ( x ,α , x ,α ) − L ,α (cid:12)(cid:12)(cid:1)(cid:13)(cid:13) < ε. (4.2) Remark 4.1.1.
The convergence in theorem 4.3 will be denoted aslim x →∞ f ( x ) = L, (4.3)and the convergence in theorem 4.4 will be denoted aslim x →−∞ f ( x ) = L. (4.4) Examples 4.1.1.
1. lim x →∞ x − − x = − α ∈ (0 , α − cut (cid:20) [2 , x ,α , x ,α ] − [1 , , − [ x ,α , x ,α ] (cid:21) = (cid:20) min i,j =1 , (cid:26) x i,α x j,α − − x i,α x j,α (cid:27) , max i,j =1 , (cid:26) x i,α x j,α − − x i,α x j,α (cid:27)(cid:21) of 2 x − − x has the limit " lim x ,α →∞ x ,α →∞ min i,j =1 , (cid:26) x i,α x j,α − − x i,α x j,α (cid:27) , lim x ,α →∞ x ,α →∞ max i,j =1 , (cid:26) x i,α x j,α − − x i,α x j,α (cid:27) = " lim x ,α →∞ x ,α →∞ min i,j =1 , (cid:26) − /x i,α x j,α − /x i,α x j,α (cid:27) , lim x ,α →∞ x ,α →∞ max i,j =1 , (cid:26) − /x i,α x j,α − /x i,α x j,α (cid:27) = [ − , − .
13. lim x →∞ x = 0 = lim x →−∞ x because by resolution principle, for all α ∈ (0 , ε >
0, there exist α − cuts [ K ,α , K ,α ] of K > x ,α , x ,α ) > ( K ,α , K ,α ) ⇒ (cid:13)(cid:13)(cid:13)(cid:13)(cid:18)(cid:12)(cid:12)(cid:12)(cid:12) x ,α (cid:12)(cid:12)(cid:12)(cid:12) , (cid:12)(cid:12)(cid:12)(cid:12) x ,α (cid:12)(cid:12)(cid:12)(cid:12)(cid:19)(cid:13)(cid:13)(cid:13)(cid:13) < (cid:13)(cid:13)(cid:13)(cid:13)(cid:18)(cid:12)(cid:12)(cid:12)(cid:12) K ,α (cid:12)(cid:12)(cid:12)(cid:12) , (cid:12)(cid:12)(cid:12)(cid:12) K ,α (cid:12)(cid:12)(cid:12)(cid:12)(cid:19)(cid:13)(cid:13)(cid:13)(cid:13) < ε and for all ε ′ >
0, there exist α − cuts (cid:2) K ′ ,α , K ′ ,α (cid:3) of K ′ > x ,α , x ,α ) < (cid:0) − K ′ ,α , − K ′ ,α (cid:1) ⇒ (cid:13)(cid:13)(cid:13)(cid:13)(cid:18)(cid:12)(cid:12)(cid:12)(cid:12) x ,α (cid:12)(cid:12)(cid:12)(cid:12) , (cid:12)(cid:12)(cid:12)(cid:12) x ,α (cid:12)(cid:12)(cid:12)(cid:12)(cid:19)(cid:13)(cid:13)(cid:13)(cid:13) < (cid:13)(cid:13)(cid:13)(cid:13)(cid:13) (cid:12)(cid:12)(cid:12)(cid:12)(cid:12) K ′ ,α (cid:12)(cid:12)(cid:12)(cid:12)(cid:12) , (cid:12)(cid:12)(cid:12)(cid:12)(cid:12) K ′ ,α (cid:12)(cid:12)(cid:12)(cid:12)(cid:12)!(cid:13)(cid:13)(cid:13)(cid:13)(cid:13) < ε ′ . Concept of fuzzy limit of fuzzy function at infinity will be given here through the following theorems whoseproofs are similar to proofs of theorems 2.1 and 2.2.
Theorem 4.2.1.
Let f : E ⊂ R → R be a fuzzy function and ( a, ∞ ) ⊆ E for some a ∈ R . Then f ( x )converges to ∞ as x approaches a if for all α ∈ (0 , α − cut of f ( x ) converge to ∞ as thebounds of α − cut of x approach the bounds of α − cut of a . Theorem 4.2.2.
Let f : E ⊂ R → R be a fuzzy function and ( −∞ , a ) ⊆ E for some a ∈ R . Then f ( x )converges to −∞ as x approaches a if for all α ∈ (0 , α − cut of f ( x ) converge to −∞ as thebounds of α − cut of x approach the bounds of α − cut of a . Remark 4.2.1.
The convergence in theorem 4.2.1 will be denoted aslim x → a f ( x ) = ∞ , (4.5)and the convergence in theorem 4.2.2 will be denoted aslim x → a f ( x ) = −∞ . (4.6) Examples 4.2.1.
1. lim x → x = ∞ because by the resolution principle, for all α ∈ (0 , α − cuts [ M ,α , M ,α ] of M ∈ R such that 0 < k ( x ,α , x ,α ) k < δ ⇒ f ( x ,α , x ,α ) > /δ ⇒ δ = 1 /M ,α , < k ( x ,α , x ,α ) k < δ ⇒ f ( x ,α , x ,α ) > /δ ⇒ δ = 1 /M ,α , where f ( x ,α , x ,α ) = min (cid:8) /x ,α , /x ,α x ,α , /x ,α (cid:9) , f ( x ,α , x ,α ) = max (cid:8) /x ,α , /x ,α x ,α , /x ,α (cid:9) .
2. lim x → − x +22 x − x +1 = −∞ because by the resolution principle, for all α ∈ (0 , α − cuts[ M ,α , M ,α ] of M < < k ( | x ,α − | , | x ,α − | ) k < δ ⇒ f ( x ,α , x ,α ) = min i,j =1 , (cid:26) x i,α + 22 x i,α x j,α − x i,α + 1 (cid:27) < M ,α , < k ( | x ,α − | , | x ,α − | ) k < δ ⇒ f ( x ,α , x ,α ) = max i,j =1 , (cid:26) x i,α + 22 x i,α x j,α − x i,α + 1 (cid:27) < M ,α , where 2 x i,α x j,α − x i,α + 1 is negative and converges to 0 as ( x ,α , x ,α ) approaches to (1 ,
1) fromthe left. Therefore, choosing δ i ∈ (0 , , i = 1 , − δ , − δ ) < ( x ,α , x ,α ) < (1 ,
1) and(1 − δ , − δ ) < ( x ,α , x ,α ) < (1 ,
1) imply 2 /M < x i,α x j,α − x i,α +1 and 2 /M < x i,α x j,α − x i,α +1respectively. Since (0 , < ( x ,α , x ,α ) < (1 ,
1) imply (2 , < ( x ,α + 2 , x ,α + 2) < (3 , Conclusion
Concept of Limit of function can be generalized to fuzzy limit of fuzzy functions. Basic properties that rulethe classical concept of limit of function can be also generalized and proved in light of fuzzy logic and fuzzysets. Future works like fuzzy continuity, fuzzy derivation and fuzzy integration of fuzzy functions and theirproperties will be considered depending on concept of fuzzy limit of fuzzy function and its basic properties.
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