The Mean Value Theorem in the Context of Generalized Approach to Differentiability

: The article is a natural continuation of the systematic research of the properties of the generalized concept of differentiability for functions with a domain X ⊂ R n that is not necessarily open, at points that allow a neighbourhood ray in the domain. In the new context, the well-known Lagrange’s mean value theorem for scalar functions is stated and proved, even for the case when the differential is not unique at all points of the observed segment in the domain. Likewise, it has been proven that its variant is valid for vector functions as well. Additionally, the paper provides a proof of the generalization of the mean value theorem for continuous scalar functions continuously differentiable in the interior of a compact domain.


Introduction
The first recorded version of the theorem we know today as Lagrange's mean value theorem dates back to the 12th century, and the first proof of its special case known as Rolle's theorem was given by Rolle in the late 17th century, albeit only for polynomials and without the tools of differential calculus.After him, great mathematicians throughout history dealt with the theorem: Maclaurin, Euler, Lagrange, Drobisch, Liouville, Serret, and the first proof of the theorem expressed in the form we know today was given by Cauchy in 1823.Actually, Cauchy proved a generalization of the mean value theorem, called Cauchy's mean value theorem (Theorem 5.12 in [1]).After him, mathematicians' interest in the mean value theorem did not stop, and many versions of its proofs as well as proofs of its various variants can be found in the literature (see [2]).The significance, practicality and application of Lagrange's theorem can be found, for example, in [3].
This article presents further results of research on the differentiability of functions in the context of the generalized approach to differentiability introduced in [4,5].The Lagrange's mean value theorem and its generalizations for scalar and vector functions are studied and proved to be valid in this new concept.
In this article, we prove that the statement about the existence of a tangent to the graph of a function (with appropriate properties) parallel to the observed secant also applies to a real function of several variables whose domain is not necessarily open, nor is the differential necessarily unique at all points of the observed segment in the domain.
The main contribution of this paper is the proof of the generalization of the mean value theorem for functions with a compact domain.Namely, it has been proven that if the points of the graph of a scalar function (with appropriate properties) corresponding to the edge of the compact domain are in the same (hyper)plane, there exists a tangential plane to the graph of the function parallel to that plane.
It is known that the mean value theorem in this form is not valid for vector functions with an open domain [1], so it is not valid in this new context of the generalized approach to differentiability either.However, a certain, very useful, generalization of it applies to vector functions and the statement and the proof of the corresponding theorem are given in the article.

Preliminaries
In this chapter, we will briefly state the definitions of the terms that appear in the rest of the article, and which can be found in more detail in the articles [4,5] to which this article is a natural continuation.
For X ⊂ R n and some P 0 ∈ X, the linearization space at P 0 with respect to X, denoted by Σ X,P 0 , is a linear hull of the set A point P 0 ∈ X admits a neighbourhood ray in X if the set ∆ X,P 0 is not empty.If in addition the point P 0 has at least one neighborhood U ⊂ X such that P 0 P ⊂ U holds for every P ∈ U, we say that P 0 admits a raylike neighbourhood in X.
In the sequel follows a generalized definition of the differentiability of a function, at a point which does not have to be an interior point of the function's domain.Let us recall that the interior of a set is the largest open set (in Euclidean topology) contained in it, denoted Int.
We say that a function f : X → R m is differentiable at the point P 0 ∈ X which allows the neighbourhood ray in it if there exists a linear operator Λ : R n → R m such that lim H→0 H∈∆ X,P 0 In this case, we call the linear operator Λ the differential of the function f at the point P 0 .Every linear operator B : R n → R m that coincides with Λ on the subspace Σ X,P 0 is the differential of the function f at the point P 0 .Therefore, the differential of a function at a point does not have to be unique, but all differentials coincide on Σ X,P 0 .In the case when Σ X,P 0 = R n , the differential is unique, and we denote it by d f (P 0 ).
The function f is continuously differentiable if it is differentiable on X, if at each point P ∈ X the differential is unique and if the mapping d f : X → Hom(R n , R m ) is continuous.Here, Hom(R n , R m ) is a normed vector space of all linear operators with the operator norm and d f associates with each point P the linear operator d f (P).If also every point P ∈ X admits a raylike neighbourhood in X, f is a function of the class C 1 .As commented in [5], we remark that if the domain of the function is open, the previous definition of differentiability agrees with the well-known definition of this term.Also, in that case, the notion of "being of the class C 1 " coincides with the notion of continuous differentiability.
To continue, we will need some well-known theorems of mathematical analysis.
Theorem 1 (Weierstrass theorem, pp.89-90 in [6]).Let X be a nonempty compact space and f : X → R a continuous function.Then, f attains its minimum and maximum value, each at least once.If in addition X is connected, then the image f (X) is a segment [min f (X), max f (X)].
Using Fermat's theorem, we obtain the following generalization.

Theorem 3.
Let Ω ⊂ R n be an open set, P 0 ∈ Ω and let f : Ω → R be a function differentiable at P 0 .If f reaches its minimum or maximum value at P 0 , then d f (P 0 ) = 0.
Proof.Recall that a Euclidean space R n with a metrizable topology induced by the metric d 2 has the same topological structure as a space R n with a topology induced by the metric d ∞ .Therefore, since Ω is an open set, there is a ball in the metric d ∞ around P 0 of radius r contained in Ω on which the function in P 0 has a minimum or maximum value.Now, the function φ i : which is differentiable at x 0 i .Hence, Moreover, the real function of the real variable φ i has its minimum or maximum value at x 0 i , and by Theorem 2 Theorem 4 (Rolle's theorem, 5.10 in [1]).
The statement of Rolle's theorem can be generalized to a statement related to real functions of several variables.Theorem 5. Let K be a compact subset of R n with Int K = ∅ and f : K → R a continuous function differentiable on Ω = Int K.If a restriction f Fr K of the function f on the boundary Fr K is constant, then there exists a point P 0 ∈ Ω such that d f (P 0 ) = 0.
Proof.According to the Weierstrass Theorem 1, the function f reaches on K its maximum value M and its minimum value m.If M = m, then f is a constant function and d f (P) = 0, for every P ∈ Ω.If m < M, then one of the following inequalities must hold: M = c or m = c, where c = f (P) for every P ∈ Fr K.In both cases, there is a point P 0 ∈ Ω at which f , and consequently f Ω reaches its minimum or maximum value.By the Fermat Theorem 3, we have d f Ω (P 0 ) = 0, and consequently d f (P 0 ) = 0.
Finally, let us recall the mean value theorem.

The Mean Value Theorem in the Context of Generalized Approach to Differentiability
The analogue of Lagrange's theorem is also valid in the case of a real function of several variables, even when the domain of the function is not necessarily open, nor is the differential necessarily unique at all points of the observed segment in the domain.
Theorem 7. Let X ⊂ R n , P 0 ∈ X, H ∈ R n \{0} and let P 0 P 0 + H be a line segment contained in X.If f : X → R is a differentiable function at every point P ∈ P 0 P 0 + H, then there exists θ ∈ 0, 1 such that f where A θ : R n → R is any differential of f at the point P 0 + θH.
Proof.Let us observe a function χ : [0, 1] → R, χ(t) = f (P 0 + tH), which is a composition of χ 1 : R → R n , χ 1 (t) = P 0 + tH, and the function f .According to Theorem 6 in [4], the function χ is differentiable and where A t : R n → R is any differential of f at the point P 0 + tH, for every t ∈ [0, 1], and Ξ is a linear operator Ξ : R → R n , Ξ(h) = hH.By equating the matrix representatives of the linear operator d χ (t) and A t • Ξ, we obtain Now, by the Lagrange mean value Theorem 6 there exists θ ∈ 0, 1 such that By Corollary 1 in [4], we know that when the linearization space at the point admitting a neighbourhood ray in the domain X ⊂ R n of the function is equal to R n , the differential of the function at that point is unique if it exists.Therefore, the following corollary holds.
Corollary 1.Let X ⊂ R n , P 0 ∈ X, H ∈ R n \{0} and let P 0 P 0 + H be a line segment contained in X.If f : X → R is a differentiable function at every point P ∈ P 0 P 0 + H around which the linearization space is equal to R n , then there exists θ ∈ 0, 1 such that When the domain Ω ⊂ R n of the function f is open, according to Proposition 2 in [4] every point P 0 ∈ Ω admits a neighbourhood ray in Ω in the direction of any vector.Consequently, if V 1 , V 2 , . . ., V n are linearly independent vectors and if f is differentiable in P 0 ∈ Ω, by Theorem 8 in [4], we know that f has the derivatives at P 0 in the direction of V 1 , V 2 , . . . ,V n and that for any choice of vector As a result, we have the following corollary.

Corollary 2.
Let Ω ⊂ R n be an open set, V 1 , . . ., V n linearly independent vectors, P 0 ∈ Ω, differentiable function at every point P ∈ P 0 P 0 + H, then there exists θ ∈ 0, 1 such that Interpreting the previous theorem geometrically, we can say that there is always a point P 0 + θH on the line segment P 0 P 0 + H such that the hyperplane parallel to the tangential plane to the graph of f in (P 0 + θH, f (P 0 + θH)) containing (P 0 , f (P 0 )) also passes through the point (P 0 + H, f (P 0 + H)), i.e., the normal of that plane is perpendicular to the secant through the points (P 0 , f (P 0 )) and (P 0 + H, f (P 0 + H)).

The Mean Value Theorem for Scalar Functions
The following generalization of Lagrange's theorem gives an even nicer geometric interpretation and represents its full analogue, which is why we will call it the mean value theorem for scalar functions.We will show that it holds: if all points (P, f (P)), P ∈ Fr K, of the graph of a continuous function f : K ⊂ R n → R on the compact K, Ω = Int K = ∅, continuously differentiable at Ω, are located on the same (hyper)plane ρ, then there is a point P 0 ∈ Ω such that the tangential plane to the graph of the function f in the point (P 0 , f (P 0 )) is parallel to the plane ρ.Theorem 8. Let K be a compact subset of R n with Int K = ∅ and f : K → R a continuous function of the class C 1 on Ω = Int K.If there are real numbers a 0 , a 1 , . . ., a n such that f (P) = a 0 + a 1 x 1 + • • • + a n x n for every point P = (x 1 , . . ., x n ) ∈ Fr K, then there is P 0 ∈ Ω such that grad f (P 0 ) = (a 1 , . . ., a n ).

Proof. Let us define a function
are continuous, for every i = 1, . . ., n, therefore φ is differentiable.Since φ(P) = 0 for every P ∈ Fr K, the function φ satisfies the conditions of the Theorem 5, so there is a point P 0 = (x 0 1 , . . ., x 0 n ) ∈ Ω such that dφ(P 0 ) = 0.This means that ∂ i (P 0 ) = a i for every i = 1, . . ., n, which proves the statement of the theorem.

The Mean Value Theorem for Vector Functions
It is well known that, although it makes sense, the statement of the Theorem 7 does not hold for vector functions (see [1]).However, a certain, very useful generalization applies to vector functions.Theorem 9. Let X ⊂ R n , P 0 ∈ X, H ∈ R n \{0}, P 0 P 0 + H ⊂ X, Σ X,P = R n for all P ∈ P 0 P 0 + H, and let f : X → R m be a differentiable function at every P ∈ P

Remark 1. It is clear from the context what the norm sign
represents: if it acts on a vector, then it represents the Euclidean norm on R n or R m and if it acts on the linear operator d f (P), then we use it for the operator norm on Hom(R n , R m ).
Proof.The inequality obviously holds if Let us define a function χ Let σ : R m → R be a linear functional determined by the vector Obviously, g(t) = Q | f (P 0 + tH) .Furthermore, according to Theorem 6 in [4], g is differentiable on 0, 1 and, according to Proposition 3 in [4] and the Lagrange mean value Theorem 6, there is θ ∈ 0, 1 such that On the other hand, using the properties of the scalar product, we obtain Now, according to Schwarz inequality, we have Finally, since Q = 1, we conclude An example of the application of the previous theorem can be found in the proof of the converse of the statement that every constant function is differentiable and that its differential at every point is the zero-operator.It is immediately apparent that the converse statement is not always valid.For example, the function f : is not a constant mapping even though it is differentiable at every point and its differential at every point is the zero-operator.However, the reverse statement will hold with some additional assumptions.
Theorem 10.Let X ⊂ R n be a set in which every two points are connected by a polygonal path.Let f : X → R m be a differentiable function for which the linearization space around each point of the domain is equal to R n .If d f (P) = 0 for every point P ∈ X, then f is a constant mapping.
Proof.Let P 0 ∈ X be some chosen point and P ∈ X an arbitrary point of the domain of the function f .It is enough to prove that f (P 0 ) = f (P).By this assumption, there is a polygonal path in X that connects points P 0 and P, i.e., there are points P 1 , . . ., P n such that P 0 P 1 ∪ P 1 P 2 ∪ • • • ∪ P n P is contained in X.Also, by assumption, thus by applying the mean value theorem for vector functions, we obtain Therefore, f (P 0 ) = f (P 1 ).In the same way we obtain f (P i ) = f (P i+1 ), i = 1, . . ., n − 1 and f (P n ) = f (P), so f (P 0 ) = f (P).
Since every region is connected by polygonal paths, the following corollary follows directly from the previous theorem.

Corollary 3.
Let Ω ⊂ R n be a region and f : Ω → R m a differentiable function such that d f (P) = 0 for every point P ∈ Ω.Then, f is a constant mapping.Corollary 4. Let X ⊂ R n , P 0 ∈ X, H ∈ R n \{0}, P 0 P 0 + H ⊂ X and let f : X → R m be a function of the class C 1 .Then, Proof.Since the line segment P 0 P 0 + H is compact and the norm and function d f are continuous, according to the Weierstrass theorem, the set d f (P) | P ∈ P 0 P 0 + H is bounded, so the statement follows from the previous theorem.
Furthermore, since the function f : [x 0 , x 0 + h] → R is continuous on the compact, according to the Weierstrass theorem, it reaches its maximum at some point x ∈ [x 0 , x 0 + h].Now, the statement follows from the previous corollary.

Conclusions
In the articles [4,5], the differentiability of vector functions is defined not only on open sets in R n , but much more widely, i.e., wherever the concept of differentiability and linearization makes sense, and even at points where the differential of the function is not unique.As a consequence of this definition, interesting phenomena appear such as functions that are differentiable but not continuous, or functions that are differentiable at points where there are no partial derivatives, and their role is taken over by derivatives along linearly independent vectors.Some well-known theorems such as the inverse function theorem and the composition theorem (see, e.g., [7,8]) can only now be expressed in their full generality in this new context, as was conducted in [4].Also, the notion of continuous differentiability can now be introduced in a natural way as a property of the continuity of a function that maps linear operators to points of the domain of the function, and the usual way, through the continuity of partial derivatives, becomes an operational characterization of that concept.
This article provides some new results in the framework of generalized differentiality that are applicable in all areas where standard differentiable calculus can be applied.For some other generalizations of the concept of differentiability, see, for example, [9] or [10].
This article shows that the statement about the existence of a tangent to the graph of a function parallel to the observed secant is also valid for a real function of several variables whose domain is not necessarily open, nor is the differential necessarily unique at all points of the observed segment in the domain.In the case when the differential is not unique, the statement is valid for any differential of the function at the abscissa of the point of contact of the tangent.It is proved that the variant of the theorem for vector functions is also valid in the same context.Additionally, the paper gives a proof of an interesting generalization of the mean value theorem for scalar functions with a compact domain that has not been seen in the literature so far.
The future scope of this paper would be to continue researching the properties of the generalized concept of differentiability.