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Generalized Differentiability of Continuous Functions

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Generalized Differentiability of Continuous Functions fractal and fractional Article Generalized Differentiability of Continuous Functions Dimiter Prodanov 1,2 1 Environment, Health and Safety, IMEC vzw, Kapeldreef 75, 3001 Leuven, Belgium; [email protected] 2 MMSDP, IICT, Bulgarian Academy of Sciences, Acad. G. Bonchev St., Block 25A, 1113 Sofia, Bulgaria Received: 19 October 2020; Accepted: 3 December 2020; Published: 10 December 2020 Abstract: Many physical phenomena give rise to mathematical models in terms of fractal, non-differentiable functions. The paper introduces a broad generalization of the derivative in terms of the maximal modulus of continuity of the primitive function. These derivatives are called indicial derivatives. As an application, the indicial derivatives are used to characterize the nowhere monotonous functions. Furthermore, the non-differentiability set of such derivatives is proven to be of measure zero. As a second application, the indicial derivative is used in the proof of the Lebesgue differentiation theorem. Finally, the connection with the fractional velocities is demonstrated. Keywords: fractals; Smith–Volterra–Cantor set; Cantor functions; differentiable functions; Lipschitz (Hölder) classes MSC: 26A30; 26A16 1. Introduction From a certain perspective, a derivative can be interpreted as an idealization of the linear approximation. Moreover, the derivatives are indispensable tools for physical modelling. In classical physics, the second law of Newton tacitly assumes that the velocity is a differentiable function of time. This ensures mathematical modelling in terms of differential equations, and hence (almost everywhere) differentiable functions. Ampere even tried to prove that all functions are almost everywhere differentiable [1]. Now we know that this attempt was doomed to fail. Various non-differentiable functions were constructed in the 19th century and regarded with a mixture of wonder and horror (historical survey in [2]). For example, Charles Hermite wrote “I turn with terror and horror from this lamentable scourge of functions with no derivatives”, while Henri Poincaré denounced these functions as “an outrage against common sense”. The interest in fractal and non-differentiable functions was rekindled with the works of Mandelbrot in fractals [3]. Moreover, many physical phenomena give rise to models in terms of fractal, non-differentiable functions. Such an opposition leads to the question whether the ordinary (i.e., Cauchy) derivatives can be generalized locally in a way that can describe such irregular objects. This is certainly achievable for a class of singular functions, such as the De Rham’s function, which can be characterized locally in terms of fractional velocity [4,5]. Purely mathematically, the derivatives can be generalized in several ways. Derivatives can be defined in the usual way as limits of difference quotients on the accumulation sets of points [6], (Chaper 3, p. 105). This approach can be applied also to functions defined on perfect fractal sets as demonstrated in [7–9]. Having physical applications in mind, it is logical to restrict the functions of study to continuous ones. Such functions will be allowed to vary everywhere on the real line. So this condition makes the setting very different from the approach of Pravate, Gangal and Yang. Fractal Fract. 2020, 4, 56; doi:10.3390/fractalfract4040056 www.mdpi.com/journal/fractalfract Fractal Fract. 2020, 4, 56 2 of 17 A very broad generalization of derivatives can be achieved if the linear, Lipschitz, growth condition is replaced by a non-linear one. From this perspective, the growth condition can be taken as the modulus of continuity. Such notion is by necessity limited. However, it provides a framework which allows one to identify if a certain property of derivatives carries over to more particular constructs, for example in the setting of fractional velocities [10], or it is special for the linear growth condition. The paper is organized as follows. Section2 introduces some general definitions and notations. Section3 introduces the oscillation of functions. Section4 introduces indicial derivatives. Section5 gives the construction of a singular function used to clarify the Lebesgue differentiation theorem. Section6 characterizes the continuity sets of derivatives. Section7 characterizes the fractional velocities in terms of the indicator derivatives. 2. General Definitions and Conventions The term variable denotes an indefinite number taken from a predefined set, usually the real numbers. Sets are denoted by capital letters, while variables taking values in sets are denoted by lowercase. The term function denotes a mapping from one number to another and the action of the function is denoted as f (x) = y. Implicitly the mapping acts on the real numbers: f : R 7! R. If a statement of a function f fulfils a certain predicate with argument A (i.e., Pred[A]) the following short-hand notation will be used f =∼ Pred[A]. The co-domain of the function f : X 7! Y is denoted as f [X] = Y. The term operator denotes the mapping from a functional expression to functional expression. The term functional denotes the mapping from a functional expression to a number. Square brackets are used for the arguments of operators and functionals, while round brackets are used for the arguments of functions. The term Cauchy sequence will always be interpreted as a null sequence. Everywhere, e will be considered as a small positive variable. The asymptotic small O notation will be used as introduced in [5]. Definition 1. Define the parametrized difference operators acting on the function f (x) as + De [ f ](x) := f (x + e) − f (x), − De [ f ](x) := f (x) − f (x − e) for the variable e > 0. The two operators are referred to as forward difference and backward difference operators, respectively. Definition 2 (Anonymous function notation). The notation for the pair m :: e will be interpreted as the implication that if Left-Hand Side (LHS) is fixed then the Right-Hand Side (RHS) is fixed by the value chosen on the left, i.e., as an anonymous functional dependency e = e(m). 3. The Discontinuity Sets of Functions In order to state the main result of the section we introduce the auxiliary concept of function oscillation. Definition 3 (Oscillation). Define the oscillation of the function f in the interval J = [a, b] as osc f := sup f − inf f J J J Fractal Fract. 2020, 4, 56 3 of 17 Definition 4 (Directed Oscillation). Define the directed oscillations as (i) the forward oscillation: + osce [ f ] (x) := sup f − inf f , I = [x, x + e] [x,x+e] [x,x+e] and the backward oscillation: − osce [ f ] (x) := sup f − inf f , I = [x − e, x] [x−e,x] [x−e,x] Finally, define the limits, if such exist as finite numbers, as osc+[ f ](x) := lim sup f − inf f , I = [x, x + e] jIj!0 I I osc−[ f ](x) := lim sup f − inf f , I = [x − e, x] jIj!0 I I where j · j denotes the length of the interval, according to previously introduced notation [5,11]. The lemma was stated in [11]: Lemma 1 (Oscillation lemma). Consider the function f : X 7! Y ⊆ R . Suppose that I+ = [x, x + e] ⊆ X, I− = [x − e, x] ⊆ X, respectively. If osc+[ f ](x) = 0 then f is right-continuous at x. Conversely, if f is right-continuous at x then osc+[ f ](x) = 0. If osc−[ f ](x) = 0 then f is left-continuous at x. Conversely, if f is left-continuous at x then osc−[ f ](x) = 0. That is, ± lim osce [ f ] (x) = 0 () lim f (x ± e) = f (x) e!0 e!0 Then the negation of the statement is also true. Corollary 1. The following two statements are equivalent ± lim osce [ f ] (x) > 0 () lim f (x ± e) 6= f (x) e!0 e!0 Theorem 1 (Disconnected discontinuity set). Consider a bounded function F defined on the compact interval I. Then its set of discontinuity D[F, I] is totally disconnected in I. Proof. Consider a decreasing collection of closed nested intervals from a partition of the interval n I1 = [x, x + h] (h > 0, not necessarily small); that is fIk = [x, x + ak]gk=1, Ik+1 ⊂ Ik ⊂ ... ⊂ I1. Since the case when the function is locally constant is trivial we consider only two cases: increasing and decreasing. Let En = In \ In−1 and define Dn := sup F − inf F I In n Dn−1 := sup F − inf F I − In−1 n 1 DE := sup F − inf F E En n Increasing case: Suppose that F is increasing in In−1 then Fractal Fract. 2020, 4, 56 4 of 17 Dn−1 − Dn = sup F − inf F − sup F + inf F = sup F − inf F − inf F + inf F = DE = osc f I − In I − En I − En In−1 n 1 In En n 1 n 1 Decreasing case: Suppose that F is decreasing in In−1 then Dn−1 − Dn = sup F − inf F − sup F + sup F = DE = osc f E E In n In En n On the other hand, osc f = Dn and osc f = Dn−1. If F is continuous on the opening of In−1, In In−1 in limit lim osc f − osc f = 0 ) lim osc f = 0 n!¥ In In−1 jEnj!0 En and D[F, I] n fxg = Æ. On the other hand, F can be discontinuous on the boundary points fx, x + an−1g, which are disconnected. Suppose then that both endpoints are points of discontinuity. Hence, fx, x + an−1g ⊆ D[F, I]. If F is discontinuous on the opening of In−1, we take In and proceed in the same way. Therefore, D[F, I] 6= Æ is a union of totally disconnected sets and hence it is totally disconnected. The disconnected discontinuity theorem is used as the main tool characterizing the continuity sets of derivatives.
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