Nondifferentiability and fractional differentiability on timescales
Nondifferentiability and fractional differentiability on timescales
Mehdi Nategh 0
Abdolali Neamaty 0
Bahram Agheli 0
Mathematics Subject Classification 0
0 B. Agheli Department of Mathematics, Qaemshahr Branch, Islamic Azad University , Qaemshahr , Iran
This work deals with concepts of nondifferentiability and a noninteger order differential on timescales. Through an investigation of a local noninteger order derivative on timescales, a mean value theorem (a fractional analog of the mean value theorem on timescales) is presented. Then, by illustrating a vanishing property of this derivative, its objectivity is discussed. As a firsthand result, the potentials and capability of this fractional derivative connected to nonsmooth analysis, including nondifferentiable paths and a class of selfsimilar fractals, are stated. It is stated that the noninteger order derivative never vanishes almost everywhere. It has been shown that with the help of changing the order of differentiability on a qtimescale, the nondifferentiability disappears.

26E70 · 26A33 · 28A80
1 Introduction
Calculus on timescales originally was proposed in S. Hilger’s Ph.D. thesis, which aimed to unify the differential
and difference calculus [
15, 16
]. Timescales, as closed subsets of the real line, include diverse patterns such as
noncountable selfsimilar sets, primes and closed intervals as well. Although its fabric is one dimensional, up
to an appropriate transformation, it is potentially capable of including higher dimensional patterns (see Sect.
2.1). Despite the complexity of the diverse scattering patterns in timescales, dynamic equations and inequalities
on timescales are enriched by many fruitful results, and indeed research topics in this area are of great interests
[
4, 8, 9
].
Compared to the classical calculus with some of its nonlocal fractional counterparts with the same antiquity,
a few incomplete efforts aiming to present local fractional calculus [
3
] or nonlocal fractional calculus [
2, 26
]
on timescales have been done.
Differentiation and/or integration on fractals are reported in [
7, 12, 20, 27
]. In conjunction with the main
result of the present work, it is noteworthy that [27] introduces a local noninteger order calculus on Cantor
set, which (in its general topological definition) is a prototype of a famous and mostly observed pattern.
The mentioned theory in [
27
] is thrived to obtain an almost comprehensive achievement in noninteger order
dynamic equations on Cantor set in the local sense. Surprisingly, this local noninteger order theory studies a
subclass of nondifferentiable functions including Lebesgue–Cantor function.
Based on the idea in [
7
], a noninteger order derivative is in accordance with the Hausdorff dimension of
a compact fractal. A rigorous mathematical integration theory on fractals can be found in [
12
], in which the
concept of a generalized Riemann integral based on the staircase function is introduced. Moreover, for a given
compact fractal F and a function f : F → R, it proposes a derivative at t ∈ F by (provided the limit exists)
f (s) − f (t )
lim
s→t ∗ SFα (s) − ∗ SFα (t )
s∈F
where ∗ SFα is the staircase function corresponding to F , and 0 < α < 1 is the ∗γ dimension of F [
12
].
A timescale approach toward the differentiation theory on fractals is suggested in [
3
]. Indeed, [
3
] proposes
a noninteger order differentiation on timescales that contains compact fractals as well. In this approach, an
implicit assumption is that all timescales are subject to the local noninteger order differentiation theory. In
other words, for a given function defined on an arbitrary timescale, the differential of that function is not
equivalent to zero necessarily.
As it will be discussed in this paper, in the abovementioned proposal, all those timescales whose set
of dense points are nonempty must be excluded from the theory since the derivative of a smooth function
vanishes at the dense points. To overcome the lack of applicability in the presence of this vanishing property,
one may take nondifferentiable functions into account and this supposition leads us to the main result of the
present work.
This work is organized as follows:
Section 2 is devoted to preliminaries on timescales with and αderivative. In Sect. 2.1, a class of
selfsimilar fractals as compact timescales are studied. Noninteger order mean value on timescales is stated in
Sect. 2.2. The mentioned theorem is a breakthrough in both integer and noninteger order differentiation and
integration theory on timescales, whereas it is utilized to obtain the main background for integration theory [
5
].
In Sect. 3, we obtain the vanishing property of αderivative for smooth functions in the sense that f (α) vanishes
at rightdense points. Then, utilizing the noninteger order mean value theorem, we obtain the main result for
nondifferentiable and αdifferentiable functions. Section 4 is devoted to some examples of nondifferentiable
functions.
In accordance with a classical nowhere differentiability and the classical Weierstrass function, a class of
nondifferentiable functions on a qtimescale is introduced. Then, the αdifferentiability is represented. The
specific series form of those functions is inspired by the trigonometric series form of their classical archetype,
namely the Weierstrass function. Also, the qversion of a nondifferentiable function in the above sense appears
as a qintegral, which is a summation consisting of a finite number of terms.
2 Preliminary concepts
For comprehensive discussions on the calculus and dynamic equations on timescales, see [
5
].
Timescales are closed subsets of the real line. Some of the prevalent examples are Z, hZ (h > 0), Primes,
2Z and R.
Three basic functions denoted by σ , ρ and μ, respectively, are defined by
σ (t ) := inf {s  s > t },
ρ(t ) := sup {s  s < t },
μ(t ) := σ (t ) − t.
A rightscattered point t ∈ T is a point for which we have μ(t ) > 0, and it is called rightdense when μ(t ) = 0.
Tκ is defined by T\{t0}, where t0 is the leftscattered maximum. A function f : T → R is called regulated
when its leftsided limits exist at leftdense points and its rightsided limits exist at rightdense points. It is
clear that regulated functions are bounded on bounded subsets of T. A function f : Tκ → R is said to be
rdcontinuous, if it is continuous at rightdense points, and the leftsided limits on leftdense points exist. The
notation Crd (T) or simply Crd is served as the set of rdcontinuous functions.
2.1 Selfsimilar compact timescales
The notion of timescales and noninteger order dynamic may be supported by fractal geometry. In view of the
prevalence of fractal patterns in nature, some inspiring studies, specially in biology, are reported [
1,13,14
].
The present subsection is inspired by Fig. 1, [
14
].
Suppose S ⊂ RN and X = { (an)n  an ∈ Z p } ( p ∈ Primes), X1 ⊂ X be nonvoided and assume that there
is a one to one correspondence between S and X1. Making use of pbased expansion of numbers, [
−1, 1
]/∼
R
consists of equivalence classes t ∈ Z , with t1 ∼ t2 iff an − bn = 0 mod p, where t1 = n apnn and t2 = n bpnn .
Define T : X → RZ by T (an)n = n apnn , then (X, d) is an infinite dimensional vector space over Z p, where
d((an)n, (bn)n ) = T (an − bn)n. It is clear that T = T X1 is a timescale provided (X1, d) is closed in (X, d).
For the next statement, members of s ∈ S are denoted by the same notations as their counterparts in X ,
i.e., s = {an}, so d(s1, s2) for s1, s2 ∈ S is defined by d({an}, {bn}), where {an} and {bn} are correspondent X
members for s1 and s2.
Proposition 2.1 Let g : S → S be a function and suppose
d(g(s1), g(s2)) = λdα(s1, s2)
for some α ∈ [
0, 1
] and 0 < λ < 1. If S is correspondent to X1 ⊂ X , then a similar identity holds for
T ◦ g ◦ T −1 : T → T, where T = T (X1) ⊂ RZ .
Proof Let t1 =
n∈N0 n∈N0
in S. Let {a˜n} and {b˜n} be correspondent to g(s1) and g(s2), then
apnn and t2 =
bpnn , then, T −1(t1) and T −1(t2) are X correspondent preimages of t1, t2
T ◦ g ◦ T −1(t1) − T ◦ g ◦ T −1(t1) = T (g(s1) − g(s2))
(2.1)
(2.2)
(2.3)
a˜n − b˜n
pn
=
n∈N0
= d({a˜n}, {b˜n})
= λdα({an}, {bn})
= λ
n∈N0
= λt1 − t2α.
an − bn α
pn
(2.4)
(2.5)
(2.6)
(2.7)
(2.8)
Remark 2.2 The result of Proposition 2.1 holds if we substitute the equality sign by ≤.
Let α = 1 and Eq. (2.3) with ≤ holds for {g1, g2, . . . , gn}, with 0 < λi < 1, then there exists a compact
subset S∗ ⊂ S such that S∗ = in=1 gi (S∗) (see Chapter 9, [
11
]). According to Proposition 2.1, the same
holds for {T ◦ gn1 ◦ T −1, T ◦ g2 ◦ T −1, . . . , T ◦ gn ◦ T −1}, i.e., there exists a compact subset T∗ ⊂ T, for
which T∗ = i=1 T ◦ gi ◦ T −1(T∗). Suppose the open set condition holds in the sense that there exists an
open U ⊂ R2 such that in=1 gi (U ) ⊂ U with disjoint gi (U )s (accordingly, in=1 T ◦ gi ◦ T −1(V ) ⊂ V , for
(ssoemeeCohpaepnteVr 2⊂,[1RZ1]w),itthhednisbjoyinTthTeo◦regmi◦9T.3−[11(1V],)s). inI=f 1γλ=iγ =di 1m.H(S∗) denotes the Hausdorff dimension of S∗
Using the scaling property (Sec 2.1, [
11
]) for 0 < α < 1, we obtain a similar result, while for the
corresponding compact invariant set S∗ when α = 1, the exponent γ should be substituted by γα and we have
n γ
i=1 λiα = 1.
2.2
Calculus and fractional calculus on timescales
Calculus on timescales begins with the following definition:
Definition 2.3 [
5
] Suppose X is a real Banach space, f : T → X is a function and t ∈ Tκ . We define
f (t ) ∈ X to be a vector—if it exists—with the property that, for any > 0, there is a neighborhood
U := (t − δ, t + δ) for some δ > 0 such that
f (σ (t )) − f (s) − (σ (t ) − s) f (t ) ≤ σ (t ) − s.
In case μ(t ) > 0 and f is continuous, then f is differentiable at t and we have
f (t ) − f (s)
t − s
and when the point t is rightdense, f is differentiable at t iff the limit
exists.
For D ⊂ Tκ , a continuous function f : T → X is called predifferentiable with region of differentiation
D, if T\D is countable and contains no rightscattered point and also f (t ) exists for all t ∈ D.
A noninteger order derivative in the local sense has been suggested in [
21
].
n
Definition 2.4 Let f be a realvalued function defined on T, t ∈ Tκ and α ∈ (0, 1] ∩ { 2m+1  n, m ∈ N}, the
real value f (α)(t ) (if it exists) is defined to be a number with the property that, for any positive , there exists
a neighborhood U ⊂ T of t of length δ so that
 f (σ (t )) − f (s) − f (α)(t )(σ (t ) − s)α ≤ σ (t ) − sα,
for all s ∈ U . We name the number f (α)(t ) fractional derivative of order α at t .
1
exists and then it will be f (α)(x ) [
3
]. Similarly, if α ∈ (0, 1] ∩ { q  q is odd } and t is leftdense. then f is
fractional differentiable of order α at t iff the limit
exists and then it will be f (α)(t ) [
3
].
Remark 2.5 Reminding the notion of measure chain which leads us to the graininess function μ, a fractional
counterpart of it can be thought of as a premeasure on a bent real line with the deflection parameter α. In this
way, a rough geometric interpretation of fractional derivative of a function at point t ∈ T of order α is a slope
of a bent line tangent to the function at t ∈ T.
Theorem 2.6 [
5
] Suppose f, g : T → R are predifferentiable functions with D. Then
 f (t ) ≤ g (t ),
∀t ∈ D
 f (s) − f (r ) ≤ g(s) − g(r ),
∀r, s ∈ T, r ≤ s.
 f (s) − f (r ) ≤
sup
t∈U κ ∩D
 f (t ) s − r .
Corollary 2.7 [
5
] Suppose U = [r, s] or [s, r ], where r, s ∈ T. If f : T → R is predifferentiable functions
with D, then the following inequality holds:
A noninteger order counterpart of Theorem 2.6 is stated below:
Theorem 2.8 (Fractional Mean Value) Let f, g : T → R be αpredifferentiable functions with D. If
Besides the αdifferentiation theory in [
3
], an indefinite integral together with a number of firsthand properties
are derived. However, some of those properties with their applicability and domain of validity have been
discussed in [
21
].
Similar to Eqs. (2.6) and (2.7), if f is continuous at t and t is rightscattered, then f is differentiable of
order α at t with [
3
]
1
Let α ∈ (0, 1] ∩ { q  q be odd }. If t is rightdense, then f is fractional differentiable of order α at t iff the limit
Proof The proof is based on the induction principle ([
5
], Theorem 1.7) and here it is given with a modification
to the proof of Theorem 1.67 in [
5
].
Let r, s ∈ T with r ≤ s. Since T \ D is countable, we denote it by {tn  n ∈ N}. Suppose the following
statement
S(t ) : there exists a continuous and bounded function M : Tκ → R such that
 f (t ) − f (r ) ≤ g(t ) − g(r ) +
The trivial case S(r ) is satisfied with any bounded and continuous M . Let t be rightscattered, then t ∈ D and
assume that S(t ) holds. We have
 f (σ (t )) − f (r ) =  f (t ) + μα (t ) f (α)(t ) − f (r )
f (α)(t ) =
f (σ (t )) − f (t )
μα (t )
.
f (α)(t ) :=
lim
s−→t+
f (t ) − f (s)
(t − s)α
f (α)(t ) :=
lim
s−→t−
f (t ) − f (s)
(t − s)α
 f (α)(t ) ≤ g(α)(t ),
∀t ∈ D,
 f (s) − f (r ) ≤ g(s) − g(r ),
∀r, s ∈ T, r ≤ s.
implies
then
(2.9)
(2.10)
(2.11)
(2.12)
(2.13)
(2.14)
(2.15)
(2.16)
(2.17)
This shows that S(σ (t )) holds with M (σ (t )) = M (t ).
Now, let t be rightdense with t = s. First, suppose t ∈ D. Since f and g are predifferentiable, then there
exists a neighborhood U of t with
 f (t ) − f (τ ) − (τ − t )α f (α)(t ) ≤ 2 τ − t α
g(τ ) − g(t ) − g(α)(t )(t − τ )α ≤ 2 τ − t α
∀ τ ∈ U
∀ τ ∈ U
This shows that, for τ ∈ U ∩ (t, ∞), we have
 f (τ ) − f (r ) ≤  f (τ ) − f (t ) +  f (t ) − f (r )
≤  f (α)(t )(τ − t )α + 2 (τ − t )α +  f (t ) − f (r )
≤ g(α)(t )(τ − t )α + 2 (τ − t )α +  f (t ) − f (r )
≤ g(τ ) − g(t ) + (τ − t )α + g(t ) − g(r ) +
Therefore, the statement S(τ ) for τ ∈ U ∩ (t, ∞) holds with M (τ ) = 1 + M (t ).
Assume the other case, namely t ∈ T \ D or t = tm for some m. Since every predifferentiable function is
continuous, there exists a neighborhood U of t such that for every τ ∈ U , we have
Hence,
 f (τ ) − f (r ) ≤  f (τ ) − f (t ) +  f (t ) − f (r )
 f (τ ) − f (t ) ≤ 2 2−m ,
g(τ ) − g(t ) ≥ 2 2−m .
2−m + g(t ) − g(r ) +
M (t )(t − r )α +
2−n
2−n .
(2.18)
(2.19)
(2.20)
(2.21)
(2.22)
The last inequality holds since t = tm < τ . Therefore, S(τ ) holds for all τ ∈ U ∩ (t, ∞) with M (τ ) = M (t ).
Now, let t be a leftdense point and suppose S(τ ) holds for τ < t . Thus, we have
τl→imt−  f (τ ) − f (r ) ≤ lim
τ →t−
≤ lim
τ →t−
g(τ ) − g(r ) +
M (τ )(τ − r )α +
g(τ ) − g(r ) +
M (τ )(τ − r )α +
tn<τ
tn<t
2−n
2−n
and S(t ) follows from continuity of f , g and M with M (t ) =
deductions prove the assertion.
lim M (τ ). We conclude that the above
τ →t−
where (y − x )α = u. Here, the uniform convergence of the series and uniform continuity of the family (u n−αα )n
give the result.
Remark 3.2 The above result fails in the of lack of differentiability: Suppose T = { tn  n ∈ N } ∪ {t } with
tn → t and let f (t ) = (tn − t )α, then f (α)(t ) = 1 while t is dense.
Evidently, for a function satisfying f (s) = O((s − t )α) when s → t implies f (α)(t ) = 0 up to the dense
points, but such asymptotic condition ruins the differentiability.
Proposition 3.3 Suppose that Tκn = ∅ where Tκk+1 = (Tκk )κ . Assume that f : T → R is ntimes
continuously differentiable. If x ∈ Tκn is a dense point, then f (α)(x ) = 0.
Proof According to [
5
], Theorem 1.111 (Taylor’s Formula), since g1(y, x )α = (y − x )α, then
3 Main results
The content of this section evolves around the objectivity of the αderivative and it aims to show that the class
of αdifferentiable functions with nonzero αderivative is rather small in the sense of smoothness. It is assumed
that T is bounded and functions are continuous. Since the concentration is on the rightdense points, for the
sake of objectivity, assume that the set of rightdense points is nonempty.
Proposition 3.1 Let T = [a, b] and assume that f is analytic on a neighborhood of a point x ∈ [a, b]. Then
f (α) ≡ 0.
Proof By the definition, we have
(3.1)
(3.2)
(3.3)
Indeed, from the definition of gk , that is g0(y, x ) = 1 and gk+1(y, x ) = xy gk (σ (t ), x ) t , for t ∈ {t ∈ T  x ≤
t < y}, by induction we obtain
xy gk (σ (t ), x ) t
g1(y, x )α
(y − x )k+1
≤ g1(y, x )α
= (y − x )k−α+1, k ∈ N,
since σ (t ) ≤ y and it completes the proof.
In favor of Theorem 2.8, one can easily obtain:
Proposition 3.4 Let f be a predifferentiable function with D and t ∈ D be a (left or right)dense point. If
f is bounded on D, then f (α)(t ) = 0.
Corollary 3.5 Suppose f ∈ C 1(Tκ ), then f (α) vanishes at rightdense points.
Proof Assume f (α)(t ) = 0 for some rightdense t . From Proposition 3.3, there exists a sequence (tn)n ⊂ Tκ
with f (tn) → ∞. Let t0 = lim tn, then f is not differentiable on t0. It yields t0 ∈/ Tκ and therefore
t0 = max T and it contradicts the rightdense property. This proves the assertion.
gx0 (x ) =
f (x ) − f (x0)
(x − x0)α
we have gx0 ∈ Hβ−α(T) ⊂ C (T) and gx0 Hβ−α(T) ≤ K f Hβ (T), where K does not depend on f e.g.,([
22
]
section 1.1). Since gx0 is continuous on T, its limit exists at x0 provided x0 ∈ E and one can infer that f is
locally Hölder continuous. Also, by compactness of T and continuity of gt with respect to t , the family {gt }t∈T
is uniform bounded and it proves f ∈ Hα(T).
Parts 2 and 3 are clear.
4) Suppose f ∈ h0α and let g(t ) = supt∈T\E  f (t ) t , then we have
(3.4)
(3.5)
(3.6)
Theorem 3.8 (Rademacher’s Theorem, [
24
], Corollary 11.7) If
then f is differentiable almost everywhere.
⊂ Rn is open and f :
→ Rm is Lipschitz,
As we already observed, there is no hope of looking inside the class of smooth functions for finding
nontrivial αdifferentiable functions. If N D([a, b]) stands for the set of all nowheredifferentiable functions over
[a, b], then almost every function in C [a, b] belongs to N D[a, b] ([
25
], Theorem 4.8); hence, because of the
prevalence of N D[a, b] in C [a, b], it seems to be reasonable to regard the αderivative in connection with
nonsmooth analysis.
For the purpose of the following statement, let Z ( f (α)) be the set of zeros of f (α).
Lemma 3.9 Let α, β ∈ R+ with α < β. Then
21.. HLiβ((TT))∪⊂hHαα(T),
+ ⊂ Hα(T),
3. Li (T) ∩ hα+ = ∅,
4. Suppose T does not contain leftdense points. Assume that f
f ∈ Li (T).
Proof Let x0 ∈ T and assume that f ∈ Hβ (T), then for the function gx0 defined by
is bounded on T \ E and f ∈ h0α, then
Definition 3.6 Let E denote the set of rightdense points in T. The set of all αpredifferentiable functions
f : T → R for which f (α) is bounded and nonzero on E ∩ D, is denoted by hα+. Also, h0α stands for the set of
all αpredifferentiable functions with f (α) ≡ 0 on E ∩ D and Li (T) denotes the space of Lipschitz functions.
0 ⊂ Hα(T), in which Hα(T) is the the space of Ho¨lder continuous functions on T, which is a
Clearly, hα , hα
Banach sp+ace with the norm
f Hα(T) =
f
+ sup
x,y∈T
f (x ) − f (y)
(x − y)α
,
in which . is the sup norm.
The following statement is immediate.
Proposition 3.7 Li (T) ⊂ h0α.
g(α)(t ) =
0
supt∈T\E  f (t ) μ1−α(t ) t ∈/ E
t ∈ E
and therefore  f (α)(t ) ≤ g(α)(t ) (t ∈ Tκ ) holds. The result is obvious from Theorem 2.8 since g ∈ Li (T).
Theorem 3.10 Suppose f is almost nowhere differentiable, but αpredifferentiable with D.
1. There exists at least one point t for which f (α)(t ) = 0.
2. If f (α) is continuous, then it does not vanish a.e.
Proof Part 1 is an immediate consequence of Theorem 2.8.
2) Let A = Z ( f (α)) (zeros of f (α)), then A ⊂ D is a timescale. By Theorem 2.8, f is constant on A and
therefore A has full measure.
4 Nondifferentiability vs αdifferentiability: examples
Test Example 4.1 It is well known that the Weierstrass function defined by the following trigonometric series
f (x ) =
∞
n=1
b−nα cos bn x ,
0 < α < 1, b > 1
is nondifferentiable and belongs to Hα . Fractal geometric study of the Weierstrass function (as a monofractal)
and its generalization has an interesting literature, both in theory and in applications [
6, 17, 18, 28
].
Although in the present work, we have studied the local fractional derivative, nonlocal based fractional
studies of f are reported [
23
]. Indeed, there is a direct relationship between the order of Grunwald–Letnikov
derivative of f and its Hausdorff dimension [
23
].
Now, let { n }n be such that n → 0, then the function
∞
n=1
g(x ) =
n b−nα cos bn x
sup g(x ) − g(y) = o(x − yα ),
has the following property:
n∈N
We define f : T → R by
which leads to g(α) ≡ 0 (Chapter 2, Section 3, [
29
]). Then, we conclude that there exists a nonvoided set
upon which g is differentiable (Fig. 1).
Test Example 4.2 (αDifferentiability on qtimescales)
√ N
Suppose q ∈]0, 52−1 [, and define T1 = {0} ∪ q = {q n  n ∈ N}, T1n = {0} ∪ q Nn+1 , T = {1} ∪ T1 ∪
(q n + Tn1 ). Let l : R → R be a periodic function defined by l(t ) = t  (t ∈ [
−1, 1
]) and l(t ) = l(t + q1 ).
which has a q integral representation of the form (see [
10, 19
] for theory of integration and differentiation on
qtimescales.)
It is clear that T1 ⊂ T consists of dense points of T, and f is
t = qk for some k ∈ N, and let t j = q j + qk , in which j > k. Obviously, {t j } ⊂]qk , qk−1[ since q ∈]0,
Following that, it is easy to obtain
differentiable on T\T1. Let t ∈ T1, then
√
f (t ) = jl→im∞
f (q j + qk ) − f (qk )
q j
= jl→im∞
j + 1 +
( j − k)qk
q j
= +∞
(since h(qk q−n) = qk−n when n ≤ k and otherwise h(qk q−n) = 0). The same result holds for
g(t ) =
qnαl(q−nt ),
0 < α < 1, t ∈ T,
qnα.qk−n −
q j
qn(α−1)
= +∞.
k
n=0 qnα.qk−n
while we have
f (t ) = jl→im∞
= jl→im∞
j
n=0
j
n=0
qnα.q j−n +
qn(α−1)
qk
+ q j
n∈N0
j
n=0
j
n=k+1
where T˜ = {0, 1} ∪ qαN
of order 0 < β < 1 on {0} ∪n∈qNαqNα0:n
f (α)(t ) = jl→im∞
f (q j + +qk ) − f (+qk )
q jβ
= jl→im∞ (qα−1 − 1)q jβ
qα−1
Note that with the help of q˜ = qα, Eq. (4.7) has a qintegral representation below
g(t ) = 1 − qα
qα
qαN∪{0,1}
l
t
1
τ α
dq τ,
t ∈ T˜,
+ qαNn+1 (Fig. 2). It can be verified that g is not fractional differentiable
q jα − q j−α+1 + qk− j+ jα − qkα
= ∞.
For the purpose of fractional differentiability, we need to modify the previous definitions: Let l : R → R
be a differentiable function that vanishes outside [
0, 1
]. We can define the corresponding function h by
n∈N0
The summation in Eq. (4.11) is finite again, while l vanishes outside of [
0, 1
], so for t = qk , we have
l(qαk−n ) = 0 for αk < n, where . is the floor function. The function h is fractional differentiable of order
α, since for t = qk we have
h(α)(t ) = jl→im∞
h(q j + qk ) − h(qk )
q jα
1
= jl→im∞ q jα
αk
n=0
qnαl(qkα−n )
k
n=0
(4.6)
(4.7)
(4.8)
(4.9)
(4.10)
(4.11)
1
Fig. 3 h(t ) as a function of T˜ with l(t ) = t (1 − t ) for t ∈ [
0, 1
], otherwise l(t ) = 0, and α = 3
=
=
αk
n=0
αk
n=0
q nα lim
j→∞
l(q jα−n + q kα−n ) − l(q kα−n )
q jα
q (α+1)nl (q αk−n ).
Note that one can obtain a similar result by considering T˜ and defining
n∈N0
Apparently, h(t ) in either of the above definitions is not differentiable, but αdifferentiable on T and
equivalently, h as a function on T˜ is differentiable (see Fig. 3).
Remark 4.3 As it has been represented through the examples above, the discussed local noninteger order
derivative and derivative are equal up to a change of scale, i.e., changing q N0 with q αN0 in Example 4.2.
Interestingly, the transformation t −→ t α preserves the scattering property of two discussed timescales, that
is, μ(t ) = 1 −qq t for t ∈ T and μ(t ) = 1 −qαqα t for t ∈ T˜. The same invariance holds for T = Z under the shift
transformation t −→ t + α. In general, if a given timescale is invariant under a transformation (a change of
scale as a function of α), then we would be able to use and α derivative interchangeably in the sense of T
and T˜ and their corresponding function h(t ).
5 Conclusion
In this work, an objectivity of the local αderivative on timescales has been discussed. Using the fractional mean
value theorem, it is shown that, compared to the smooth functions, which have trivial αderivative, nowhere
differentiable functions, which satisfy fractional differentiability, possess almost everywhere nontriviality of
the fractional derivative of order α. As it is shown in Example 4.1, the discussed local derivative has some
potentials to study dynamics on selfsimilar fractals. The concepts of nowhere differentiability up to dense
points and αdifferentiability on timescales is also studied, and almost everywhere nonvanishing property
of a fractional order derivative is stated. Making use of the series representation of qintegrals, a class of
nondifferentiable, but αdifferentiable on a qtimescale is introduced.
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(4.12)
(4.13)
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