A Mathematical Journal
Vol. 7, No 3, (1 - 13). December 2005.

Fuzzy Taylor Formulae

George A. Anastassiou
Department of Mathematical Sciences University of Memphis

Memphis, TN 38152 U.S.A.
ganastss@memphis.edu

ABSTRACT
We produce Fuzzy Taylor formulae with integral remainder in the univariate

and multivariate cases, analogs of the real setting.

RESUMEN
Se presentan versiones Fuzzy análogas a las reales de fórmulas de Taylor con

resto integral en el caso univariado y multivariado.

Key words and phrases: Fuzzy Taylor formula, Fuzzy–Riemann integral
remainder, H-fuzzy derivative, fuzzy real analysis.

2000 AMS Subj. Class.: 26E50.

1 Background

We need the following

Definition A (see [10]). Let µ : R → [0, 1] with the following properties.

(i) is normal, i.e., ∃x0 ∈ R; µ(x0) = 1.



2 George A. Anastassiou
7, 3(2005)

(ii) µ(λx + (1 − λ)y) ≥ min{µ(x), µ(y)}, ∀x, y ∈ R, ∀λ ∈ [0, 1] (µ is called a convex
fuzzy subset).

(iii) µ is upper semicontinuous on R, i.e., ∀x0 ∈ R and ∀ε > 0, ∃ neighborhood
V (x0): µ(x) ≤ µ(x0) + ε, ∀x ∈ V (x0).

(iv) The set supp(µ) is compact in R (where supp(µ) := {x ∈ R; µ(x) > 0}).

We call µ a fuzzy real number. Denote the set of all µ with RF.
E.g., X{x0} ∈ RF, for any x0 ∈ R, where X{x0} is the characteristic function at x0.
For 0 < r ≤ 1 and µ ∈ RF define [µ]r := {x ∈ R: µ(x) ≥ r} and

[µ]0 := {x ∈ R : µ(x) > 0}.

Then it is well known that for each r ∈ [0, 1], [µ]r is a closed and bounded interval of
R. For u, v ∈ RF and λ ∈ R, we define uniquely the sum u ⊕ v and the product λ � u
by

[u ⊕ v]r = [u]r + [v]r, [λ � u]r = λ[u]r, ∀r ∈ [0, 1],

where [u]r + [v]r means the usual addition of two intervals (as subsets of R) and λ[u]r
means the usual product between a scalar and a subset of R (see, e.g., [10]). Notice
1 � u = u and it holds u ⊕ v = v ⊕ u, λ � u = u � λ. If 0 ≤ r1 ≤ r2 ≤ 1 then
[u]r2 ⊆ [u]r1 . Actually [u]r = [u(r)− , u

(r)
+ ], where u

(r)
− ≤ u

(r)
+ , u

(r)
− , u

(r)
+ ∈ R, ∀r ∈ [0, 1].

For λ > 0 one has λu(r)± = (λ � u)
(r)
± , respectively.

Define
D : RF × RF → R+

by
D(u, v) := sup

r∈[0,1]
max{|u(r)− − v

(r)
− |, |u

(r)
+ − v

(r)
+ |},

where [v]r = [v(r)− , v
(r)
+ ]; u, v ∈ RF. We have that D is a metric on RF. Then (RF, D)

is a complete metric space, see [10], with the properties

D(u ⊕ w, v ⊕ w) = D(u, v), ∀u, v, w ∈ RF,
D(k � u, k � v) = |k|D(u, v), ∀u, v ∈ RF, ∀k ∈ R,
D(u ⊕ v, w ⊕ e) ≤ D(u, w) + D(v, e), ∀u, v, w, e ∈ RF.

Let f, g : R → RF be fuzzy number valued functions. The distance between f, g is
defined by

D∗(f, g) := sup
x∈R

D(f (x), g(x)).

On RF we define a partial order by “≤”: u, v ∈ RF, u ≤ v iff u
(r)
− ≤ v

(r)
− and

u
(r)
+ ≤ v

(r)
+ , ∀r ∈ [0, 1].

We mention



7, 3(2005)
Fuzzy Taylor Formulae 3

Lemma 2.2 ([5]). For any a, b ∈ R : a, b ≥ 0 and any u ∈ RF we have

D(a � u, b � u) ≤ |a − b| · D(u, õ),

where õ ∈ RF is defined by õ := X{0}.

Lemma 4.1 ([5]).

(i) If we denote õ := X{0}, then õ ∈ RF is the neutral element with respect to ⊕,
i.e., u ⊕ õ = õ ⊕ u = u, ∀u ∈ RF.

(ii) With respect to õ, none of u ∈ RF, u 6= õ has opposite in RF.

(iii) Let a, b ∈ R : a · b ≥ 0, and any u ∈ RF, we have (a + b) � u = a � u ⊕ b � u.
For general a, b ∈ R, the above property is fale.

(iv) For any λ ∈ R and any u, v ∈ RF, we have λ � (u ⊕ v) = λ � u ⊕ λ � v.

(v) For any λ, µ ∈ R and u ∈ RF, we have λ � (µ � u) = (λ · µ) � u.

(vi) If we denote ‖u‖F := D(u, õ), ∀u ∈ RF, then ‖·‖F has the properties of a usual
norm on RF, i.e.,

‖u‖F = 0 iff u = õ, ‖λ � u‖F = |λ| · ‖u‖F,
‖u ⊕ v‖F ≤ ‖u‖F + ‖v‖F, ‖u‖F − ‖v‖F ≤ D(u, v).

Notice that (RF, ⊕, �) is not a linear space over R, and consequently (RF, ‖ ·‖F)
is not a normed space.

We need

Definition B (see [10]). Let x, y ∈ RF. If there exists a z ∈ RF such that x = y + z,
then we call z the H-difference of x and y, denoted by z := x − y.

Definition 3.3 ([10]). Let T := [x0, x0 + β] ⊂ R, with β > 0. A function f : T → RF
is H-differentiable at x ∈ T if there exists a f′(x) ∈ RF such that the limits (with
respect to metric D)

lim
h→0+

f (x + h) − f (x)
h

, lim
h→0+

f (x) − f (x − h)
h

exist and are equal to f′(x). We call f′ the derivative or H-derivative of f at x. If
f is H-differentiable at any x ∈ T , we call f differentiable or H-differentiable and it
has H-derivative over T the function f′.

The last definition was given first by M. Puri and D. Ralescu [9].
We need also a particular case of the Fuzzy Henstock integral (δ(x) = δ

2
) introduced

in [10], Definition 2.1.
That is,



4 George A. Anastassiou
7, 3(2005)

Definition 13.14 ([6], p. 644). Let f : [a, b] → RF. We say that f is Fuzzy-Riemann
integrable to I ∈ RF if for any ε > 0, there exists δ > 0 such that for any division
P = {[u, v]; ξ} of [a, b] with the norms ∆(P ) < δ, we have

D

(∑
P

∗(v − u) � f (ξ), I

)
< ε,

where
∑∗ denotes the fuzzy summation. We choose to write

I := (F R)
∫ b

a

f (x)dx.

We also call an f as above (F R)-integrable.
We mention

Lemma 1 ([3]). If f, g : [a, b] ⊆ R → RF are fuzzy continuous functions, then the
function F : [a, b] → R+ defined by F (x) := D(f (x), g(x)) is continuous on [a, b], and

D

(
(F R)

∫ b
a

f (x)dx, (F R)
∫ b

a

g(x)dx

)
≤
∫ b

a

D(f (x), g(x))dx.

Lemma 2 ([3]). Let f : [a, b] → RF fuzzy continuous (with respect to metric D), then
D(f (x), õ) ≤ M , ∀x ∈ [a, b], M > 0, that is f is fuzzy bounded. Equivalently we get
χ−M ≤ f (x) ≤ χM , ∀x ∈ [a, b].

Lemma 3 ([3]). Let f : [a, b] ⊆ R → RF be fuzzy continuous. Then

(F R)
∫ x

a

f (t)dt is a fuzzy continuous function in x ∈ [a, b].

Lemma 5 ([4]). Let f : [a, b] → RF have an existing H-fuzzy derivative f′ at c ∈ [a, b].
Then f is fuzzy continuous at c.

We need

Theorem 3.2 ([7]). Let f : [a, b] → RF be fuzzy continuous. Then (F R)
∫ b

a
f (x)dx

exists and belongs to RF, furthermore it holds[
(F R)

∫ b
a

f (x)dx

]r
=

[∫ b
a

(f )(r)− (x)dx,
∫ b

a

(f )(r)+ (x)dx

]
, ∀r ∈ [0, 1]. (1)

Clearly f (r)± : [a, b] → R are continuous functions.

We also need

Theorem 5.2 ([8]). Let f : [a, b] ⊆ R → RF be H-fuzzy differentiable. Let t ∈ [a, b],
0 ≤ r ≤ 1. (Clearly

[f (t)]r =
[
(f (t))(r)− , (f (t))

(r)
+

]
⊆ R.) (2)



7, 3(2005)
Fuzzy Taylor Formulae 5

Then (f (t))(r)± are differentiable and

[f′(t)]r =
[
((f (t))(r)− )

′, ((f (t))(r)+ )
′]. (3)

The last can be used to find f′.

Here Cn([a, b], RF), n ≥ 1 denotes the space of n-times fuzzy continuously H-
differentiable functions from [a, b] ⊆ R into RF. By above Theorem 5.2 of [8] for
f ∈ Cn([a, b], RF) we obtain

[f (i)(t)]r =
[
((f (t))(r)− )

(i), ((f (t))(r)+ )
(i)
]
, (4)

for i = 0, 1, 2, . . . , n and in particular we have

(f (i)± )
(r) = (f (r)± )

(i), ∀r ∈ [0, 1]. (5)

Definition 1. Let a1, a2, b1, b2 ∈ R such that a1 ≤ b1 and a2 ≤ b2. Then we define

[a1, b1] + [a2, b2] = [a1 + a2, b1 + b2]. (6)

Let a, b ∈ R such that a ≤ b and k ∈ R, then we define,

if k ≥ 0, k[a, b] = [ka, kb],
if k < 0, k[a, b] = [kb, ka]. (7)

Here we use

Lemma 1. Let f : [a, b] → RF be fuzzy continuous and let g : [a, b] → R+ be continu-
ous. Then f (x) � g(x) is fuzzy continuous function ∀x ∈ [a, b].

Proof. The same as of Lemma 2 ([1]), using Lemma 2 of [3].

2 Main Results

We present the following fuzzy Taylor theorem in one dimension.

Theorem 1. Let f ∈ Cn([a, b], RF), n ≥ 1, [α, β] ⊆ [a, b] ⊆ R. Then

f (β) = f (α) ⊕ f′(α) � (β − α) ⊕ ··· ⊕ f (n−1)(α) �
(β − α)n−1

(n − 1)!

⊕
1

(n − 1)!
� (F R)

∫ β
α

(β − t)n−1 � f (n)(t) dt. (8)

The integral remainder is a fuzzy continuous function in β.

Proof. Let r ∈ [0, 1]. We have here [f (β)]r = [f (r)− (β), f
(r)
+ (β)], and by Theorem 5.2

([8]) f (r)± is n-times continuously differentiable on [a, b]. By (5) we get

(f (i)± (α))
(r) = (f (r)± (α))

(i), all i = 0, 1, . . . , n, (9)



6 George A. Anastassiou
7, 3(2005)

and
[f (i)(α)]r =

[
(f (r)− (α))

(i), (f (r)+ (α))
(i)
]
.

Thus by Taylor’s theorem we obtain

f
(r)
± (β) = f

(r)
± (α) + (f

(r)
± (α))

′(β − α)

+ · · · + (f (r)± (α))
(n−1) (β − α)

n−1

(n − 1)!
+

1
(n − 1)!

∫ β
α

(β − t)n−1(f (r)± )
(n)(t)dt.

Furthermore by (9) we have

f
(r)
± (β) = f

(r)
± (α) + (f

′
±(α))

(r)(β − α)

+ · · · + (f (n−1)± (α)
(r) (β − α)

n−1

(n − 1)!
+

1
(n − 1)!

∫ β
α

(β − t)n−1(f (n)± )
(r)(t)dt.

Here it holds β − α ≥ 0, β − t ≥ 0 for t ∈ [α, β], and

(f (i)− (t))
(r) ≤ (f (i)+ (t))

(r), ∀t ∈ [a, b]

all i = 0, 1, . . . , n, and any r ∈ [0, 1].
We see that[

f
(r)
− (β), f

(r)
+ (β)] = [f

(r)
− (α) + (f

′
−(α))

(r)(β − α) + · · · + (f (n−1)− (α))
(r) (β − α)

n−1

(n − 1)!

+
1

(n − 1)!

∫ β
α

(β − t)n−1(f (n)− )
(r)(t)dt, , f (r)+ (α)

+ (f′+(α))
(r)(β − α) + · · · + (f (n−1)+ (α))

(r) (β − α)
n−1

(n − 1)!

+
1

(n − 1)!

∫ β
α

(β − t)n−1(f (n)+ )
(r)(t) dt

]
.

To split the above closed interval into a sum of smaller closed intervals is where we
use β − α ≥ 0. So we get

[f (β)r] = [f (r)− (β), f
(r)
+ (β)] = [f

(r)
− (α), f

(r)
+ (α)] + [(f

′
−(α))

(r), (f′+(α))
(r)](β − α)

+ · · · + [(f (n−1)− (α))(r), (f
(n−1)
+ (α))

(r)] (β−α)
n−1

(n−1)!

+ 1
(n−1)!

[∫ β
α

(β − t)n−1(f (n)− )(r)(t)dt,
∫ β

α
(β − t)n−1(f (n)+ )(r)(t)dt

]
= [f (α)]r + [f′(α)]r(β − α) + · · · + [f (n−1)(α)]r (β−α)

n−1

(n−1)!

+ 1
(n−1)!

[∫ β
α

((β − t)n−1 � f (n)(t))(r)− dt,
∫ β

α
((β − t)n−1 � f (n)(t))(r)+ dt

]
.



7, 3(2005)
Fuzzy Taylor Formulae 7

By Theorem 3.2 ([7]) we next get

[f (β)]r = [f (α)]r + [f′(α)]r(β − α) + · · · + [f (n−1)(α)]r
(β − α)n−1

(n − 1)!

+
1

(n − 1)!

[
(F R)

∫ β
α

(β − t)n−1 � f (n)(t)dt

]r
.

Finally we obtain

[f (β)]r =
[
f (α) ⊕ f′(α) � (β − α) ⊕ ··· ⊕ f (n−1)(α) �

(β − α)n−1

(n − 1)!

⊕
1

(n − 1)!
� (F R)

∫ β
α

(β − t)n−1 � f (n)(t)dt
]r

, all r ∈ [0, 1].

By Theorem 3.2 of [7] and Lemma 1 we get that the remainder of (8) is in RF, and by
Lemma 3 ([3]) is a fuzzy continuous function in β. The theorem has
been proved.

Next we present a multivariate fuzzy Taylor theorem.
We need the following multivariate fuzzy chain rule. Here the H-fuzzy partial

derivatives are defined according to the Definition 3.3 of [10], see Section 1, and the
analogous way to the real case.

Theorem 3 ([2]). Let φi : [a, b] ⊆ R → φi([a, b]) := Ii ⊆ R, i = 1, . . . , n, n ∈ N, are
strictly increasing and differentiable functions. Denote xi := xi(t) := φi(t), t ∈ [a, b],
i = 1, . . . , n. Consider U an open subset of Rn such that ×ni=1Ii ⊆ U . Consider
f : U → RF a fuzzy continuous function. Assume that fxi : U → RF, i = 1, . . . , n,
the H-fuzzy partial derivatives of f , exist and are fuzzy continuous. Call z := z(t) :=
f (x1, . . . , xn). Then dzdt exists and

dz

dt
=

n∑∗
i=1

dz

dxi
�

dxi
dt

, ∀t ∈ [a, b] (10)

where dz
dt

, dz
dxi

, i = 1, . . . , n are the H-fuzzy derivatives of f with respect to t, xi,
respectively.

The interchange of the order of H-fuzzy differentiation is needed too.

Theorem 4 ([2]). Let U be an open subset of Rn, n ∈ N, and f : U → RF be a
fuzzy continuous function. Assume that all H-fuzzy partial derivatives of f up to
order m ∈ N exist and are fuzzy continuous. Let x := (x1, . . . , xn) ∈ U . Then the
H-fuzzy mixed partial derivative of order k, Dx`1 ,...,x`k f (x) is unchanged when the
indices `1, . . . , `k are permuted. Each `i is a positive integer ≤ n. Here some or all
of `i’s can be equal. Also k = 2, . . . , m and there are nk partials of order k.

We give



8 George A. Anastassiou
7, 3(2005)

Theorem 2. Let U be an open convex subset of Rn, n ∈ N and f : U → RF be a
fuzzy continuous function. Assume that all H-fuzzy partial derivatives of f up to order
m ∈ N exist and are fuzzy continuous. Let z := (z1, . . . , zn), x0 := (x01, . . . , x0n) ∈ U
such that xi ≥ x0i, i = 1, . . . , n. Let 0 ≤ t ≤ 1, we define xi := x0i + t(zi − z0i),
i = 1, 2, . . . , n and gz(t) := f (x0 + t(z − x0)). (Clearly x0 + t(z − x0) ∈ U .) Then for
N = 1, . . . , m we obtain

g(N)z (t) =


( n∑∗

i=1

(zi − x0i) �
∂

∂xi

)N
f


(x1, x2, . . . , xn). (11)

Furthermore it holds the following fuzzy multivariate Taylor formula

f (z) = f (x0) ⊕
m−1∑∗
N=1

g
(N)
z (0)
N !

⊕ Rm(0, 1), (12)

where

Rm(0, 1) :=
1

(m − 1)!
� (F R)

∫ 1
0

(1 − s)m−1 � g(m)z (s)ds. (13)

Comment. (Explaining formula (11)). When N = n = 2 we have (zi ≥ x0i, i = 1, 2)

gz(t) = f (x01 + t(z1 − x01), x02 + t(z2 − x02)), 0 ≤ t ≤ 1.

We apply Theorems 3 and 4 of [2] repeatedly, etc. Thus we find

g′z(t) = (z1 − x01) �
∂f

∂x1
(x1, x2) ⊕ (z2 − x02) �

∂f

∂x2
(x1, x2).

Furthermore it holds

g′′z (t) = (z1 − x01)
2 �

∂2f

∂x21
(x1, x2) ⊕ 2(z1 − x01) · (z2 − x02) (14)

�
∂2f (x1, x2)

∂x1∂x2
⊕ (z2 − x02)2 �

∂2f

∂x22
(x1, x2).

When n = 2 and N = 3 we obtain

g′′′z (t) = (z1 − x01)
3 �

∂3f

∂x31
(x1, x2) ⊕ 3(z1 − x01)2(z2 − x02)

�
∂3f (x1, x2)

∂x21∂x2
⊕ 3(z1 − x01)(z2 − x02)2 ·

∂3f (x1, x2)
∂x1∂x

2
2

⊕ (z2 − x02)3 �
∂3f

∂x32
(x1, x2). (15)



7, 3(2005)
Fuzzy Taylor Formulae 9

When n = 3 and N = 2 we get (zi ≥ x0i, i = 1, 2, 3)

g′′z (t) = (z1 − x01)
2 �

∂2f

∂x21
(x1, x2, x3) ⊕ (z2 − x02)2 �

∂2f

∂x22
(x1, x2, x3)

⊕ (z3 − x03)2 �
∂2f

∂x23
(x1, x2, x3) ⊕ 2(z1 − x01)(z2 − x02)

�
∂2f (x1, x2, x3)

∂x1∂x2
⊕ 2(z2 − x02)(z3 − x03)

�
∂2f (x1, x2, x3)

∂x2∂x3
⊕ 2(z3 − x03)(z1 − x01) �

∂2f

∂x3∂x1
(x1, x2, x3), (16)

etc.

Proof of Theorem 2. Let z := (z1, . . . , zn), x0 := (x01, . . . , x0n) ∈ U , n ∈ N, such
that zi > x0i, i = 1, 2, . . . , n. We define

xi := φi(t) := x0i + t(zi − x0i), 0 ≤ t ≤ 1; i = 1, 2, . . . , n.

Thus dxi
dt

= zi − x0i > 0. Consider

Z := gz(t) := f (x0 + t(z − x0)) = f (x01 + t(z1 − x01), . . . , x0n + t(zn − x0n))
= f (φ1(t), . . . , φn(t)).

Since by assumptions f : U → RF is fuzzy continuous, also fxi exist and are fuzzy
continuous, by Theorem 3 (10) of [2] we get

dZ(x1, . . . , xn)
dt

=
n∑∗

i=1

∂Z(x1, . . . , xn)
∂xi

�
dxi
dt

=
n∑∗

i=1

∂f (x1, . . . , xn)
∂xi

� (zi − x0i).

Thus

g′z(t) =
n∑∗

i=1

∂f (x1, . . . , xn)
∂xi

� (zi − x0i).

Next we observe that

d2Z

dt2
= g′′z (t) =

d

dt

(
n∑∗

i=1

∂f (x1, . . . , xn)
∂xi

� (zi − x0i)

)

=
n∑∗

i=1

(zi − x0i) �
d

dt

(
∂f (x1, . . . , xn)

∂xi

)

=
n∑∗

i=1

(zi − x0i) �


 n∑∗

j=1

∂2f (x1, . . . , xn)
∂xj ∂xi

� (zj − x0j )




=
n∑∗

i=1

n∑∗
j=1

∂2f (x1, . . . , xn)
∂xj ∂xi

� (zi − x0i) · (zj − x0j ).



10 George A. Anastassiou
7, 3(2005)

That is

g′′z (t) =
n∑∗

i=1

n∑∗
j=1

∂2f (x1, . . . , xn)
∂xj ∂xi

� (zi − x0i) · (zj − x0j ).

The last is true by Theorem 3 (10) of [2] under the additional assumptions that fxi ;
∂2f

∂xj ∂xi
, i, j = 1, 2, . . . , n exist and are fuzzy continuous.

Working the same way we find

d3Z

dt3
= g′′′z (t) =

d

dt


 n∑∗

i=1

n∑∗
j=1

∂2f (x1, . . . , xn)
∂xj ∂xi

� (zi − x0i) · (zj − x0j )




=
n∑∗

i=1

n∑∗
j=1

(zi − x0i) · (zj − x0j )
d

dt

(
∂2f (x1, . . . , xn)

∂xj ∂xi

)

=
n∑∗

i=1

n∑∗
j=1

(zi − x0i) · (zj − x0j )

[
n∑∗

k=1

∂3f (x1, . . . , xn)
∂xk∂xj ∂xi

� (zk − x0k)

]

=
n∑∗

i=1

n∑∗
j=1

n∑∗
k=1

∂3f (x1, . . . , xn)
∂xk∂xj ∂xi

� (zi − x0i) · (zj − x0j ) · (zk − x0k).

Therefore,

g′′′z (t) =
n∑∗

i=1

n∑∗
j=1

n∑∗
k=1

∂3f (x1, . . . , xn)
∂xk∂xj ∂xi

� (zi − x0i) · (zj − x0j ) · (zk − x0k).

That last is true by Theorem 3 (10) of [2] under the additional assumptions that

∂3f (x1, . . . , xn)
∂xk∂xj ∂xi

, i, j, k = 1, . . . , n

do exist and are fuzzy continuous. Etc. In general one obtains that for
N = 1, . . . , m ∈ N,

g(N)z (t) =
n∑∗

i1=1

n∑∗
i2=1

· · ·
n∑∗

iN =1

∂N f (x1, . . . , xn)
∂xiN ∂xiN−1 · · · ∂xi1

�
N∏

r=1

(zir − x0ir ),

which by Theorem 4 of [2] is the same as (11) for the case zi > x0i, see also (14),
(15), and (16). The last is true by Theorem 3 (10) of [2] under the assumptions that
all H-partial derivatives of f up to order m exist and they are all fuzzy continuous
including f itself.

Next let tm̃ → t̃, as m̃ → +∞, tm̃, t̃ ∈ [0, 1]. Consider

xim̃ := x0i + tm̃(zi − x0i)

and
x̃i := x0i + t̃(zi − x0i), i = 1, 2, . . . , n.



7, 3(2005)
Fuzzy Taylor Formulae 11

That is
xm̃ = (x1m̃, x2m̃, . . . , xnm̃) and x̃ = (x̃1, . . . , x̃n) in U .

Then xm̃ → x̃, as m̃ → +∞. Clearly using the properties of D-metric and under the
theorem’s assumptions, we obtain that

g(N)z (t) is fuzzy continuous for N = 0, 1, . . . , m.

Then by Theorem 1, from the univariate fuzzy Taylor formula (8), we find

gz(1) = gz(0) ⊕ g′z(0) ⊕
g′′z (0)

2!
⊕ ··· ⊕

g
(m−1)
z (0)
(m − 1)!

⊕ Rm(0, 1),

where Rm(0, 1) comes from (13).
By Theorem 3.2 of [7] and Lemma 1 we get that Rm(0, 1) ∈ RF. That is we get

the multivariate fuzzy Taylor formula

f (z) = f (x0) ⊕ g′z(0) ⊕
g′′z (0)

2!
⊕ ··· ⊕

g
(m−1)
z (0)
(m − 1)!

⊕ Rm(0, 1),

when zi > x0i, i = 1, 2, . . . , n.
Finally we would like to take care of the case that some x0i = zi. Without loss of

generality we may assume that x01 = z1, and zi > x0i, i = 2, . . . , n. In this case we
define

Z̃ := g̃z(t) := f (x01, x02 + t(z2 − x02), . . . , x0n + t(zn − x0n)).

Therefore one has

g̃′z(t) =
n∑∗

i=2

∂f (x01, x2, . . . , xn)
∂xi

� (zi − x0i),

and in general we find

g̃(N)z (t) =
n∑∗

i2=2,...,iN =2

∂N f (x01, x2, . . . , xn)
∂xiN ∂xN−1 · · · ∂xi2

�
N∏

r=2

(zir − x0ir ),

for N = 1, . . . , m ∈ N. Notice that all g̃(N)z , N = 0, 1, . . . , m are fuzzy continuous and

g̃z(0) = f (x01, x02, . . . , x0n), g̃z(1) = f (x01, z2, z3, . . . , zn).

Then one can write down a fuzzy Taylor formula, as above, for g̃z. But g̃
(N)
z (t)

coincides with g(N)z (t) formula at z1 = x01 = x1. That is both Taylor formulae in
that case coincide.

At last we remark that if z = x0, then we define Z∗ := g∗z (t) := f (x0) =: c ∈ RF a
constant. Since c = c + õ, that is c − c = õ, we obtain the H-fuzzy derivative (c)′ = õ.
Consequently we have that

g∗(N)z (t) = õ, N = 1, . . . , m.



12 George A. Anastassiou
7, 3(2005)

The last coincide with the g(N)z formula, established earlier, if we apply there
z = x0. And, of course, the fuzzy Taylor formula now can be applied trivially for
g∗z . Furthermore in that case it coincides with the Taylor formula proved earlier for
gz. We have established a multivariate fuzzy Taylor formula for the case of zi ≥ x0i,
i = 1, 2, . . . , n. That is (11)–(13) are true.

Note. Theorem 2 is still valid when U is a compact convex subset of Rn such that
U ⊆ W , where W is an open subset of Rn. Now f : W → RF and it has all the
properties of f as in Theorem 2. Clearly here we take x0, z ∈ U .

Received: March 2003. Revised: July 2003.

References

[1] George A. Anastassiou, Fuzzy wavelet type operators, submitted.

[2] George A. Anastassiou, On H-fuzzy differentiation, Mathematica Balkanica,
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[3] George A. Anastassiou, Rate of convergence of fuzzy neural network oper-
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[4] George A. Anastassiou, Univariate fuzzy-random neural network approxima-
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[5] George A. Anastassiou and Sorin Gal, On a fuzzy trigonometric approxima-
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[6] S. Gal, Approximation theory in fuzzy setting. Chapter 13 , Handbook of
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7, 3(2005)
Fuzzy Taylor Formulae 13

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