AN ANALOGUE OF THE LOGARITHMIC (u, v)-DERIVATIVE AND ITS APPLICATION

We study an analogue of the logarithmic (u, v)-derivative. The last one has many interesting properties and good ways to calculate it. To show how it can be used we apply it to a model class of nowhere monotone functions that are composition of Salem function and nowhere differentiable functions.


Introduction
We are interested in continuous functions that are both singular (different from constant, but have a derivative equal to zero, almost everywhere, in terms of Lebesgue measure) and nowhere monotone (don't have any interval of monotonicity). Their theory is poor enough and is exhausted by a few separate examples. It is possible to expand the range of such objects by the superposition of singular and nowhere monotone functions. In a model example of a pair of known simple representatives of the class of singular functions and the class of nowhere monotone functions, we discuss the problems of a detailed study of differential properties of complex functions and propose a new toolkit for their study.
For q = 1/2, the function S(x) is linear and, for q = 1/2, it is singularly continuous. Its properties have been studied in [6].
is clear that these functions have non-trivial local properties, a study of which requires a use of nontraditional approaches. Note that even calculating the value of the function at a given point is not easy, let alone calculating the actual derivative. In the following four paragraphs we provide brief facts about the (u, v)-derivative and its analogue, the logarithmic (u, v)-derivative and, accordingly, an analogue of the logarithmic (u, v)-derivative, which provides the sought toolkit for these functions.

Key concepts and statements
Let P be a set of pairs (u, v) of all infinitesimal functions at zero, such, that for each pair there exists a number δ > 0 such that for is called the (u, v)-derivative of the function f at the point x 0 .
The first time the (u, v)-derivative was introduced in [7]. This concept is useful for the tasks of uncovering uncertainties and establishing the fact of singularity and nondifferentiability.
The following structures are related to the (u, v)-derivative.
It is easy to show that lim t1→t , and is equal to the fractal velocity [5] if t = 0.
• In [2], the conformable fractional derivate was defined to be the limit Given the design of the analogue of the (u, v)-derivative as the limit is oscillation of the function f on the segment with the endpoints x + u(h) and x − v(h), and a pair of functions (u, v) ∈ P + (P + contains all pairs of P satisfying the inequality u(h) · v(h) ≥ 0 in certain punctured neighborhood of zero). The usage of the analogue of the (u, v)-derivative allowed to show that there is a model class of functions containing singular functions that have unbounded variation on each segment from the domain of definition.
To simplify the study of compositions of functions, there was introduced a logarithmic (u, v)-derivative.
Let a function f and a pair of functions (u, v) ∈ P be given. Set the number (if it exists) for fixed x 0 from the domain of definition of f , which will be called the logarithmic (u, v)-derivative of the function f at the point x 0 . In [5] the fractal velocity of fractional order 0 ≤ β ≤ 1 was defined as Obviously Extending logarithmic (u, v)-derivative to vector functions we can see that the same relation is obtained with the fractal gradient defined in [1] as Let (u, v) ∈ P + . We will use the following notation: A proof of the following three propositions are based on this notation.
The following conditions hold true: The following conditions hold true: If there exists an (u, v)-neighborhood of x 0 (meaning an interval with endpoints at the points Let in some (u, v)-neighborhood of the point x 0 the function f have different signs. Then the following inequality holds: we pass to limits in the last inequalities. Theorem 1. Let for f , g be continuous functions and a pair of = m exists then the following conditions hold: Since the values of the functions f and g at the point x 0 are equal to the zero, we have that Next, without losing the generality, we will assume that (u, v) ∈ P ⊕ . Let us show that in the case of existence of Λf (x 0 ), the equality

Let us consider the case where
To get Λf (x 0 ) = Λ h h f (x 0 ) we pass to the limit in the last inequalities.

R. Y. OSAULENKO
To get Λ u v f (x) = +∞ we pass to the limit in (10).
Proof. From the definition of Λf (x 0 ) we get According to the definition let us consider .
. It is easy to see that where µ(h) = min{α(a), α(b)}. Then Let us pass to the limit in last inequalities (for h → 0), Theorem 3. Let (l n ; r n ) be a pair of infinitesimal sequences such that l n < l n+1 < x 0 < r n+1 < r n for all n ∈ N and lim n→∞ ln(rn+1−x0) ln(rn−x0) ln(x0−ln) . For Λf (x 0 ) to exist it is necessary and sufficient that the limits lim n→∞ On the other side, To get lim n→+∞ The other case is proved by similarly.
Using the conditions of the theorem we have f (x 0 ) ln (r n − l n ) .
Lemma 3. Let (l n , r n ) and l n ,r n be given pairs of infinitesimal sequences such that the following conditions are satisfied: (1) l n ≤ x 0 < r n ,l n <r n < x 0 ; (2) lim n→∞ l n = x 0 = lim n→∞ r n , lim n→∞ln = x 0 = lim n→∞rn ; (3) l n ,r n ⊂ [l n ; x 0 ] ⊂ [l n , r n ] for all n ∈ N; (4) lim n→∞ exist simultaneously.