Inhalt

Table of Contents

"Handbook of Advanced Mathematics"

INTRODUCTION

III

VII

VIII

XII

XIII

XIV

XVI

XVII

XVIII

XIX

XXI

XXII

XXIII

XXIV

XXV

XXVI

XXVII

XXVIII

XXIX

XXX

XXXI

XXXII

APOSTILLA

"Handbook of Advanced Mathematics"

"Handbook of Advanced Mathematics"

SIMONE MALACRIDA

Simone Malacrida (1977)

Engineer and writer, has worked on research, finance, energy policy and industrial plants.

This book explores much of advanced mathematics, starting from the milestone given by mathematical analysis and moving on to differential and fractal geometry, mathematical logic, algebraic topology, advanced statistics, and numerical analysis.

At the same time, comprehensive insights about differential and integral equations, functional analysis, and advanced matrix and tensor development will be provided.

With the mathematical background exposed, it will be possible to understand all the mechanisms for describing scientific knowledge expressed through a wide variety of formalisms.

ANALYTICAL INDEX

––––––––

INTRODUCTION

I – GENERAL TOPOLOGY

II - LIMITS AND CONTINUITY

III – DIFFERENTIAL CALCULATION

IV – INTEGRAL CALCULATION

V – STUDY OF FUNCTIONS WITH REAL VARIABLES

VI – ADVANCED ANALYTICAL GEOMETRY

VII – NON-EUCLIDEAN GEOMETRIES

VIII – REAL FUNCTIONS WITH MULTIPLE VARIABLES

IX – IMPLIED FUNCTIONS

X – ADVANCED VECTOR AND MATRIX MATHEMATICS

XI – DIFFERENTIAL GEOMETRY

XII – TENSORIAL MATHEMATICS

XIII – INTEGRAL CALCULUS FOR FUNCTIONS OF SEVERAL VARIABLES

XIV – DEVELOPMENTS IN SERIES

XV – COMPLEX ANALYSIS

XVI – FUNCTIONAL ANALYSIS

XVII – TRANSFORM

XVIII – DISTRIBUTIONS

XIX – ORDINARY DIFFERENTIAL EQUATIONS

XX – PARTIAL DIFFERENTIAL EQUATIONS

XXI – INTEGRAL AND INTEGRAL-DIFFERENTIAL EQUATIONS

XXII – ADVANCED ALGEBRA

XXIII – ALGEBRAIC STRUCTURES

XXIV – GALOIS THEORY

XXV – COMBINATORY GEOMETRY

XXVI – DISCRETE MATHEMATICS

XXVII – ADVANCED STATISTICS

XXVIII – STOCHASTIC PROCESSES

XXIX – NUMERICAL ANALYSIS

XXX – FRACTAL GEOMETRY

XXXI – NUMBER THEORY

XXXII – ADVANCED MATHEMATICAL LOGIC

APOSTILLA

INTRODUCTION

INTRODUCTION

In this book we will provide all the foundations of advanced mathematics, including both the great discipline of mathematical analysis and all the disparate fields that have arisen over the last two centuries, including, to mention only a few of them, geometry differential and fractal, non-Euclidean geometries, algebraic topology, functional analysis, statistics, numerical analysis and mathematical logic.

Almost all of these notions were developed after the introduction of the formalism of mathematical analysis at the end of the seventeenth century and, since then, the path of mathematics has always continued in parallel between this sector and all the other possible sub-disciplines that gradually side by side and have taken independent paths.

To fully understand what is presented in the manual, knowledge and prerequisites of elementary mathematics are necessary, which we will not report here, such as for example everything connected to trigonometry, analytic geometry, matrix mathematics, complex numbers and the main functions elementary of real variable.

All this knowledge is present in the previously published "Elementary mathematics manual" which is to be considered as preparatory to what will be explained below and which represents a kind of first volume of the entire mathematical knowledge for which this manual is instead the completion given by the second part.

On the importance of mathematics in today's society and on the various meanings of mathematics as an artificial and universal language that it describes la Natura, please refer, therefore, to the introduction of the aforementioned previous manual.

It remains to understand why mathematical analysis has introduced that watershed between elementary and advanced mathematics.

There are two areas that complement each other in this discourse.

On the one hand, only with the introduction of mathematical analysis has it been possible to describe, with a suitable formalism, the equations that govern natural phenomena, be they physical, chemical or of other extraction, for example social or economic. In other words, mathematical analysis is the main tool for building those mechanisms that allow us to predict results, to design technologies and to think about new improvements to introduce.

On the other hand, mathematical analysis possesses, within its very nature, a specific peculiarity which clearly distinguishes it from the previous elementary mathematics. This will be evident from the first chapter of this handbook, for now we limit ourselves to saying how mathematical analysis provides for local considerations, not exclusively punctual. Just the passage from punctuality to locality will allow to build a discourse of globality, going far beyond the previous knowable.

This manual does not claim to present all the possible facets of every single sector of advanced mathematics or even to expose the demonstrations of the infinite theorems that dot the mathematical analysis and other related disciplines. First of all it is not in the scope of the writing and then an exorbitant amount of pages would be necessary, which contrasts with the spirit of a manual, by its nature synthetic and compendium.

In this handbook two major themes will be taken up several times, to underline their mutual importance.

The first is given by advanced geometry, in all its forms, precisely to indicate the parallel path between mathematics and geometry that has been present since the dawn of history.

The second argument is typical of the leap introduced by mathematical analysis and is related to topology which, for reasons of understanding, we will present in several parts of the manual.

At the end of the book, topics of general interest which can disregard mathematical analysis, such as advanced algebra, statistics and numerical analysis, will be presented.

The last chapter will be devoted to advanced mathematical logic. On closer inspection, the first chapter of the aforementioned "Manual of elementary mathematics" was dedicated to elementary logic. Closing this manual of advanced mathematics, again with logic, is by no means a coincidence: the development of mathematics is internal to logical constructs that give the compass of reference to all human reasoning.

Each individual chapter can be considered as a complete field of mathematics in itself, but only by analyzing all the topics will it be possible to touch the vastness of mathematics and that is why the order of the chapters reflects a succession of knowledge in continuous progress.

I

GENERAL TOPOLOGY

The conceptual leap between elementary and advanced mathematics was evident only after the introduction of mathematical analysis. The fact that this discipline was local, and not punctual, led to the study and development of topology, understood as the study of places and spaces not only in a geometric sense, but in a much broader sense. The general topology gives the foundations of all the underlying sectors, among which we can include the algebraic topology, the differential one, the advanced one and so on.

We define topology as a collection T of subsets of a general set X for which the following three properties hold:

1) The empty set and the general set X belong to the collection T.

2) The union of an arbitrary quantity of sets belonging to T belongs to T.

3) The intersection of a finite number of sets belonging to T belongs to T.

A topological space is defined with a pair (X, T) and the sets constituting the collection T are open sets. Particular topologies can be the trivial one in which T is formed by X and the empty set and the discrete one in which T coincides with the set of parts of X. In the first topology only the empty set and X are open sets, while in the discrete one all sets are open sets. Two topologies are comparable if one of them is a subset of the other, while if one topology contains the other, the first is said to be finer than the second. The set of all topologies is partially ordered: the trivial topology is the least fine, the discrete is the finest, and all other possible topologies have intermediate fineness between these two.

In a topological space, a set I containing a point x belonging to X is called (open) neighborhood of x if there exists an open set A contained in I containing x:

A subset of a topological space is closed if its complement is open. Closed sets have three properties:

1) The union of a finite number of closed sets is a closed set.

2) The intersection of closed sets is a closed set.

3) The set X and the empty set are closed.

With these properties, a topology based on closed sets can be constructed. In general, a subset can be closed, open, both open and closed, neither open nor closed.

Said S a subset of a topological space X, x is a point of closure of S if every neighborhood (open or closed) of x contains at least one point of S.

Said S a subset of a topological space X, x is an accumulation point of S if every neighborhood (open or closed) of x contains at least one point of S different from x itself.

Each accumulation point is a closing point while vice versa is not valid. Locking points that are not accumulation points are called isolated points.

The set of all closure points of a given set is called closure and is denoted by cl(I). The closure in a set is a closed set and contains the starting set, moreover it is the intersection of all closed sets that contain the starting set and is the smallest closed set containing the starting set. These definitions go by the name of topological closure.

A set is therefore closed if and only if it coincides with its own closure.

Finally, the closure of a subset is a subset of the closure of the main set, and a closed set contains another set if and only if this set contains the closure of the second.

It goes without saying that the closure of the empty set is the empty set, that of the general set X is the general set X and in a discrete space each set is equal to its closure.

Said S a subset of a topological space X, x is an interior point of S if there exists a neighborhood (open or closed) of x contained in S.

The set of all interior points of a given set is called the interior and is denoted by int(I). The inner part is an open subset of the starting set, it is the union of all open sets contained in that set and it is the largest open set contained in that set. These definitions are referred to as the topological interior.

A set is open if and only it coincides with its interior, furthermore the interior satisfies the idempotence relation.

Finally, the interior of a subset is a subset of the interior of the main set, and an open set contains another set if and only if that set contains the interior of the second.

It goes without saying that the interior of the empty set is the empty set, that of the general set X is the general set X and in a discrete space each set is equal to its interior.

A closed subset of a topological space is said to be rare if it has no interior. A topological space is said to be of the first category if it is the union of a countable family of rare closed sets, vice versa it is said to be of the second category.

The internal part and the closure can be associated with operators that put these two concepts into a dual relationship.

The set difference between the closure and the interior is called the frontier, an element belonging to the frontier is called the frontier point. The frontier is also the intersection between the closure and its complement and is defined as the set of points such that each neighborhood contains at least one point belonging to the set and at least one point not belonging to this set.

The boundary of a set is closed. A set is closed if and only if its boundary is contained in the set while it is open if and only if its boundary is disjoint from it.

The frontier of a set is equal to the frontier of its complement, and the closing operation is simply the union of the set with its frontier. The boundary of a set is empty if and only if the set is both closed and open.

A subset of a topological space is locally closed if it satisfies at least one of the following conditions: it is open in its closure or it is open in any closed space or it is closed in any open space or if for each point of the subset there is an open neighborhood of this point such that the intersection between the neighborhood and the subset is closed in the neighborhood.

A topological space is said to be compact if from any family of open subsets of the space whose cover is given by:

one can extract a finite subset J in I such that the same covering relation holds. This is the so-called covering compactness and can also be defined by the use of closed sets.

A topological space is said to be compact by sequences if every sequence of points in the space admits a sub-sequence converging to a point in the space.

The Bolzano-Weierstrass theorem states that every infinite subset of a compact space admits at least one accumulation point.

A closed subset of a compact is a compact; the product of compact spaces is a compact as is the quotient.

The empty set and any set defined with the trivial topology are compact. A closed and bounded interval in the set of real numbers is compact. Every finite topological space is also compact, as is the closed sphere in RxR and the Cantor set (which we will discuss at length in the chapter dedicated to fractal geometry, almost at the end of the book). Infinite sets with discrete topology are not compact.

A space is said to be locally compact which, for each point, admits a basis of neighborhoods made up of compact sets.

A non-empty topological space is said to be connected if the only pair of disjoint subsets whose union is the space itself is given by the pair between the space and the empty set. Equivalently we can state that a topological space is connected if and only if the only subsets both open and closed are the empty set and the space itself.

A connected component of a space is called a connected subset not contained in any other connected subset. A space whose connected components are its points is said to be totally disconnected. The Cantor set and a set with discrete topology are totally disconnected.

The union of lines in the plane is a connected space if at least two lines are not parallel, while in the set of real numbers a subset is connected if and only if it is an interval in which each extreme can be infinite. Furthermore, the product of connected spaces is a connected space.

A topological space is said to be connected by arcs, or by paths, if for each pair of points in the space there is a continuous function (for the definition of continuity, see the next chapter) which connects them with equal value to the endpoints of the path. Every space connected by paths is connected, but not vice versa.

A space is locally connected if it has a system of connected neighborhoods. A path-connected topological space is simply connected if the path is contractible at will up to the transformation (called homotopy) in the constant path.

We define continuous function between topological spaces as a function for which the counterimage of every open set is open.

We define Hausdorff space as a topological space which satisfies the following axioms:

1) At least one neighborhood of the point containing the point itself corresponds to each point in space.

2) Given two neighborhoods of the same point, the intersection of these two neighborhoods is a neighborhood.

3) If a neighborhood of a point is a subset of a set, then this set is also a neighborhood of the point.

4) For each neighborhood of a point there exists another neighborhood of that point such that the first neighborhood is the neighborhood of any point belonging to the second neighborhood.

5) Given two distinct points there are two disjoint neighbourhoods.

In particular, the last axiom is called the Hausdorff separability axiom of topological spaces. The separability axioms of topological spaces can be generalized according to a category of successive refinements:

1) Spaces : for each pair of points there is an open space which contains one point and not the other.

2) Spaces : for each pair of points there are two open spaces such that both contain one of the two points but not the other.

3) Spaces : for each pair of points there are two open disjoints which contain them respectively. These are Hausdorff spaces.

4) Regular spaces: for each point and for each closed disjoint there exist two open disjoints which contain them respectively.

5) Spaces : if they are and regular.

6) Completely regular spaces: for every disjoint point and for every closed set there exists a continuous function with real values which is 0 in the closed set and 1 in the point.

7) Spaces : if they are and completely regular.

8) Normal spaces: for each pair of closed disjoints there are two open disjoints which contain them respectively.

9) Spaces : if they are and normal.

Open or closed subsets of a locally compact Hausdorff space are locally compact. Any compact Hausdorff space is second rate.

We recall that in topological spaces notions of elementary mathematics such as the concepts of countability or cardinality can be extended, thus defining countable sets and continuous sets.

A subset is dense in a topological space if every element of the subset belongs to the set or is accumulation point. Equivalent definitions are the following: a subset is dense if its closure is the topological space or if every non-empty open subset intersects the subset or if the complement of the subset has an empty interior or if each point of the space is the limit of a sequence contained in the subset.

Every topological space is dense in itself; rational and irrational numbers are dense in the set of real numbers. A space is separable if its dense subset is countable. A set is never dense if it is not dense in any open set.

A topological space is uniform if it has a family of subsets satisfying the following properties:

1) Every family of subsets contains the diagonal of the Cartesian product X x X.

2) Every family of subsets is closed under inclusion.

3) Every family of subsets is closed under the intersection.

4) If a neighborhood belongs to the topology then there exists a family of subsets belonging to the topology such that, if two pairs of points having a common point belong to the family of subsets, then the two disjoint points belong to the neighborhood.

5) If a neighborhood belongs to the topology then also the inversion of the neighborhood in the Cartesian product belongs to the topology.

A metric space is a topological space generated by a topology of a basis of circular neighborhoods. In metric spaces a metric is defined which associates a non-negative real number to two points in the space for which the following properties hold:

A function is said to be continuous at a point on a metric space if, for any choice of arbitrary positive quantities, the distance between this point and another point is bounded. Considering the spherical neighborhoods and the domain of the function we have:

A metric space is always uniform. In a metric space, the distance between a point and a set also holds, defined as:

This distance is zero if and only if x belongs to the closure of I. The distance between two points of two sets can be defined in the same way. Instead, it dictates the excess of one set over the other:

The Hausdorff distance is as follows:

A metric space is bounded if its closure is bounded. In a metric space x is a closure point if for every positive radius there exists a point within the space such that the distance between x and this point is less than the radius. In a metric space x is an interior point if there exists a positive radius such that the distance between x and a generic point belonging to the space is smaller than the radius.

A metric space is complete if every Cauchy sequence converges to an element of the space. A metric space is compact if and only if it is complete and totally bounded. A metric space is always dense in its completion.

We define a normed space as a metric space in which the distance is expressed by the norm:

The norm has the properties of being positive definite and homogeneous; moreover, the triangle inequality holds. In formulas we have:

A metric space in which the first relation does not hold is said to be semi-normed. It goes without saying that every regulated space is a metric (and therefore topological) space. An infinite-dimensional normed space is not locally compact.

A metric space in which the distance (and therefore the norm) are Euclidean is called Euclidean space. This space is the usual one of elementary geometry, in fact the n-dimensional distance is very reminiscent of the classic Pythagorean theorem:

Defined as a subset of an n-dimensional Euclidean space, a point x is closure if every open n-dimensional sphere centered on x contains at least one point of the subset. Similarly, a point x is interior if there is an open n-dimensional sphere centered at the point and contained in the subset.

Euclidean spaces are locally compact. The n-dimensional sphere, the line, the plane and any Euclidean space are simply connected. The Euclidean space consisting of the set of n-dimensional real numbers is a connected space. By the Heine-Borel theorem a subset of this Euclidean space is compact if and only if it is closed.

In a Euclidean space a convex set is a set in which, for each pair of points, the path connecting them is entirely contained in the set. A convex set is simply connected.

A homeomorphism between two topological spaces is a continuous, bijective function with continuous inverse. The homeomorphism relation between topological spaces is an equivalence relation. Two homeomorphic spaces have the same topological properties. Local homeomorphism occurs if the function is locally but not globally continuous. Every local homeomorphism is a continuous and open function, every bijective local homeomorphism is a homeomorphism, the composition of two local homeomorphisms is another local homeomorphism.

A diffeomorphism is a function between two topological spaces with the property of being differentiable (see below for the definition of differentiability), invertible and with differentiable inverse. The diffeomorphism is local if the function has these properties locally but not globally. A local diffeomorphism is a particular kind of local homeomorphism, so it is open.

An isomorphism is a bijective map such that both the function and its inverse are homeomorphisms. The structures are said to be isomorphic and are substantially identical. If there is also an ordering property, we speak of order isomorphism or isotonia.

A homotopy between two continuous functions defined in two topological spaces is a continuous function between the Cartesian product of a topological space and the unit interval [0,1] which associates to the zero point the value of the first continuous function and to the one point the value of the second continuous function. Homotopy is an equivalence relation, every homeomorphism is an equivalence of homotopy. Two homotopic topological spaces maintain the properties of path connection and simple connection.

A bijective function between two metric spaces is called isometry if it holds

If this relation is multiplicative for an arbitrary positive number other than one, it is called similarity. Furthermore, it is called uniformity if it is an isomorphism between uniform Euclidean spaces and it is a homeomorphism if it is an isomorphism between two topological spaces.

An n-dimensional topological manifold is a topological Hausdorff space in which every point has an open neighborhood which is homeomorphic to an open set in n-dimensional Euclidean space. The number n is called the dimension of the manifold. The topological manifolds of dimension one are the circle and the straight line, those of dimensions two are called surfaces (examples are the sphere, the torus, the Mobius strip, the Klein bottle). For three-dimensional topological manifolds the Poincaré conjecture holds true (which states that every three-dimensional topological manifold that is simply connected and closed is homeomorphic to a three-dimensional sphere), those of four dimension represent the space-time of general relativity. Topological manifolds are homeomorphic to Euclidean spaces and hence are locally compact.

A topological subspace is a subset of a topological space that inherits the topological structure of the space.

We refer to the following chapters for insights into advanced, algebraic, functional and vector topology. This first introduction to general topology was necessary to fully understand the innovations introduced by mathematical analysis, which we will discuss shortly.

II

LIMITS AND CONTINUITY

Mathematical analysis is that part of mathematics that deals with the infinite decomposition of dense objects or sets, therefore it is based on topological concepts expressed in the first chapter.

In particular, it involves two complementary and antithetical concepts, those of infinitesimal and infinite.

Starting from the topological definition of neighborhood, an infinitesimal is any small quantity, but always different from zero. This shifts the attention from a punctual vision, typical of elementary mathematics, to a local vision, which instead characterizes mathematical analysis, closely connected to the general topology.

The discussion of the infinite starts instead from the removal of the condition of existence typical of real numbers, according to which the denominator of a fraction must always be different from zero. In mathematical analysis, a number divided by zero results in infinity, whose symbol is as follows . If the signs agree, infinity is positive, if they disagree, it is negative. It should be noted that infinities (and infinitesimals) are not all the same, as we will soon see.

The introduction of mathematical analysis was made in the second half of the seventeenth century, by Newton and Leibnitz, and immediately had great physical and engineering applications. On the other hand, it was precisely mathematical analysis that allowed the overcoming of ancient questions, such as that of Zeno's paradox.

As far as we will say, and barring explicit exceptions, all the concepts expressed hereafter are valid only for separable topological spaces, in particular for Hausdorff spaces.

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Given a function defined on a subset X of the set of real numbers and a point of accumulation of this subset, we define the limit of the function as x tending to the point of accumulation a real number such that the distance between it and the value of the function at the point is an infinitesimal. In formulas:

In this case we say that the limit as x tending to the point of accumulation of the function is given by l.

Equivalently we can say that for every neighborhood of l there exists a neighborhood of the accumulation point such that the function belongs to the neighborhood of l.

We point out that the point of accumulation is not necessarily contained in the domain of the function, i.e. the local vision is totally independent of the punctual one.

We can extend the concept of limit if the real number l is infinite. In this case it is valid:

The limit is written like this:

A further extension is given by the condition in which the point of accumulation is at infinity, ie that the upper bound of the set X is infinite. In that case:

Which leads to this definition of limit:

Obviously analogous cases are valid for the negative signs of infinity and the two extensions can be combined for accumulation points at infinity and limit with infinite value.

If we consider the extended real set, ie the set of real numbers with the addition of infinities of both signs, all these concepts can be unified. The extended real set is ordered and is a topological space, having defined the neighborhoods of infinity as those sets which contain any half-line. With these premises, the following unifying notation of limit holds:

If the limit of a function is finite, the function is said to converge at the point of accumulation. If the limit is infinite, the function is said to diverge.

We define the right limit as the limit of the function in the right neighborhood of the accumulation point, the same is done for the left one and are indicated as follows:

The respective limit values are said by excess and by default and are indicated with the plus or minus sign in superscript.

Given two functions defined on non-disjoint domains and an accumulation point belonging to the intersection of the two domains, if the limits of the two functions exist and are finite, the following operations can be performed:

If one of the two limits is infinite, the following hold instead:

The limit operation is therefore characterized as a functional, ie as an application between a space of functions and a numerical set. This functional is linear and continuous.

Some operations on limits return forms of indeterminacy that we will study shortly.

The uniqueness theorem of the limit states that a function defined on an open set of the set of real numbers cannot have two distinct limits in a point of accumulation so the limit, if it exists, is unique.

The local boundedness theorem states that a function whose limit is finite at an accumulation point is bounded around that point.

For two functions defined in an open domain of the set of real numbers, the property holds that if one function is greater than the other in the neighborhood of an accumulation point, then also the limit of the first function is greater than the other.

From this assumption we can enunciate the comparison theorem, ie that a function between two others has the same limit as the first two if they converge to an identical limit.

The following notable limits are useful for solving the calculation of limits:

Some of these notable limits will be clarified when the series expansions of functions are introduced. For now we can see how this is reflected in the so-called asymptotic estimate.

The starting point of the asymptotic estimation is given by the assumption that the infinites are not all equal to each other and neither are the infinitesimals.

Given two polynomials, the one with greater degree has a more "powerful" infinity and therefore in a fraction it dominates over the other. By doing so, it can be seen that:

Where a are the numerical coefficients of the respective monomials of higher degree. In a completely opposite way we reason for the infinitesimals, i.e. an infinitesimal of higher order prevails over one of lower order, however overturning the behavior on the results.

Taking up concepts derived from sequences and introducing Landau symbols, we can define two functions and say that one is O-larger than the other if it happens that:

Instead, it is defined as o-small if it occurs:

The analogous concepts relating to infinitesimals are called big omega and small omega. If instead it happens that the two sequences have the same order of magnitude, the theta expression is used:

Thanks to the asymptotic estimation it is possible to solve the so-called forms of indetermination which are the following: division between infinitesimals or between infinitesimals, multiplication of an infinitesimal by an infinitesimal, subtraction between infinitesimals, exponentiation of an infinitesimal to infinity (or vice versa) and exponentiation of one to infinity.

Alongside the polynomial functions, the transcendental functions are classified as follows: logarithmic infinity is less powerful than any polynomial function, whatever the degree of the polynomial, while exponential infinity is more powerful than any polynomial function. Trigonometric functions such as sine and cosine, being oscillating and bounded, do not have characteristics at infinity, so much so that their limit at infinity does not exist. In formulas:

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A function with real variable is said to be continuous at a point (of accumulation) if its limit as x tending to that point coincides with the value of the function at that point.

In this case, the local vision coincides with the punctual one, even if the information contained at the local level is of a higher order. A function is said to be continuous if it is continuous at every point in its domain.

An analogous expression for continuous functions is given by making explicit the concept of limit:

Equivalently, topological concepts and neighborhoods can be used: a function between two topological spaces is continuous if the counterimage of every open set is open and it is continuous in a point if the counterimage of each neighborhood of the function is a neighborhood of the point.

In a metric space, a function is continuous if:

The constants, the identity function, the polynomials, the rational, exponential, logarithmic functions are continuous functions. So are sine, cosine and linear transformations between Euclidean spaces.

A function is said to be inferiorly (or superiorly) semicontinuous if it is continuous only on the inferior (or superior) limit. The integer function is upper semicontinuous, the Dirichlet function (which is zero at every irrational point and one at every rational point) is lower semicontinuous at every irrational point, upper at every rational point.

A function is continuous if and only if it is both lower and upper semicontinuous. An inferior semicontinuous function in a compact set has a minimum, an upper semicontinuous function in a compact set has a maximum.

A function is said to be uniformly continuous if:

An equivalent definition can be given for topological and metric spaces.

The constants, the identity function, the linear functions, sine and cosine are uniformly continuous functions, while polynomials with degree greater than one are not.

The Heine-Cantor theorem states that continuous functions on a compact set are uniformly continuous. It goes without saying that uniformly continuous functions are continuous.

The set of all continuous functions on a fixed real-valued domain represents a vector space, denoted by .

The composition of continuous functions is a continuous function just as the sum, the difference, the product and the quotient of two continuous functions are continuous functions, while the converse is not necessarily true.

Furthermore, if the function is bijective and the set is compact, the inverse function is also continuous.

If the function is continuous between topological spaces, the counterimage of an open (or closed) set is an open (or closed) set, while the image of a compact (or connected) set is a compact (or connected) set.

For continuous functions some fundamental theorems also hold.

The sign permanence theorem states that if the function is positive at a given point within its domain, then there exists a neighborhood of that point such that the function is positive at all points in the neighborhood.

The intermediate value theorem states that the function takes on all values between the value at one point in the domain and the value at another point in the domain.

Bolzano's theorem (or existence of zeros) states that given two points of the domain in which the function assumes discordant values, then there exists at least one point of the domain between the two previous points such that the function assumes a null value.

Weierstrass's theorem states that if the interval is closed and bounded, then the function has a maximum and a minimum in the interval (or it is a constant). This theorem extends to metric spaces in the case of compact sets.

A function that is not continuous at a point is said to have a point of discontinuity. Breakpoints are divided into three distinct species:

1) Discontinuity of the first kind: the right limit and the left limit exist and are finite, but they are different from each other.

2) Discontinuity of the second kind: at least one of the two limits between the right and the left are infinite or do not exist.

3) Discontinuity of the third kind: the right limit and the left limit exist, they are finite and equal to each other, but different from the value of the function at the point.

A function which has a discontinuity of the first kind is, for example, the sign function which has a point of discontinuity at zero.

The discontinuity points of the second kind occur when the function is evaluated in a neighborhood of an accumulation point which however does not belong to the domain of the function.

A discontinuity of the third kind is also said to be eliminated because it can be removed simply by redefining the value of that function at the point.

The Dirichlet function is discontinuous at every point, every point being a discontinuity of the first kind.

III

III

DIFFERENTIAL CALCULATION

Given a real function of real variable, we call the increment of the function around a given point, the following quantity:

While the increment of the independent variable is given by h. The incremental ratio is defined as the ratio of the increments:

If h is positive, we speak of right incremental ratio, if it is negative, of left incremental ratio.

The limit as h tends to zero of the incremental ratio is called the derivative and is indicated in various ways.

The first notation is that of Lagrange, the second is used in physics, the third is the notation of Cauchy-Euler, the fourth is that of Leibnitz, the last is that of Newton.

The derivative calculated in the right neighborhood is called the right derivative and the one calculated in the left neighborhood is called the left derivative. A function is differentiable at a point if and only if there are finite left and right limits of the incremental ratio and these limits are equal. A function is differentiable everywhere, or in an interval, if it is differentiable at any point, or at any point in the interval.

The function which assumes at each point the value of the derivative at that point is called a derivative function, precisely because it derives from the starting function.

The derivative of the derivative is called the second derivative and so on up to the n-th derivative which is indicated as follows:

Having used the previous notations to indicate the nth derivative.

A necessary condition for the derivability of a function is its continuity. Continuous and differentiable functions (therefore with continuous first derivative) are part of a vector space denoted by , while a function with continuous n-th derivative is part of the space and a function having infinite continuous derivatives is part of the space (these functions are called harmonics or smooth). It goes without saying that the following relationship holds:

In essence they are all vector subspaces of the more general vector space.

When calculating derivatives, the following rules apply:

The basic derivatives are the following:

The exponential, sine, cosine, tangent, and cotangent functions are harmonic functions.

The prime differential of a function is given by:

The successive differentials are calculated in the same way considering the successive derivatives.

Fermat's theorem on stationary points states that a continuous and derivable function at an accumulation point of an open domain has zero derivative if the point is stationary.

Rolle's theorem states that a continuous function in a closed interval and differentiable in the same interval, but open, having equal values at the ends admits at least one point inside the interval in which the derivative vanishes (therefore admits at least one stationary point).

Lagrange's theorem states that a continuous function in a closed interval and differentiable in the same interval, but open, admits at least one point inside this interval in which the following relation holds:

Where a and b are the boundary points of the interval.

Cauchy's theorem states that two continuous functions in a closed interval and differentiable in the same but open interval admit at least one point inside this interval in which the following relation holds:

Where the function g(x) has different values at the extremes and derivative always different from zero within the considered interval.

From these theorems derives the theorem of the constant function which states that a continuous function in a closed interval and differentiable in the same interval, but open, is a constant if and only if its derivative is zero everywhere in its interval.

Furthermore, the sign of the first derivative of a function is strictly connected to its monotonicity. A continuous function in a closed interval and differentiable in the same interval, but open, with always positive derivative within this interval means that it is always increasing in this interval. If, on the other hand, the derivative is always negative, it means that it is always decreasing.

The sign of the second derivative is instead related to the convexity of the function. A continuous function in a closed interval and derivable in the same interval, but open, having a definite second derivative is convex if and only if its second derivative is always positive. If its second derivative is always negative, it is concave.

As we will see, the properties of the derivatives will be fundamental for the study of functions of real variables, in particular for determining the stationary points of the first derivative (maximum, minimum or inflection with horizontal tangent) and of the second derivative (inflection with oblique tangent). Differential calculus, in addition to being useful for the study of functions, provides a rule, called Hopital's, for the resolution of some forms of indeterminacy of limits. Given two continuous functions in a closed interval and differentiable in the same interval, but open, in the case in which there is a form of indeterminacy of the type zero divided by zero or infinity divided by infinity, the following rule holds:

Where obviously the denominators are well defined in the range, except at the maximum at the point of accumulation. In essence, this theorem allows us to solve the forms of indeterminacy by applying the derivative to the numerator and denominator until we obtain a finite limit. This theorem can be applied iteratively for successive derivatives.

The derivative in a point also has a remarkable geometric meaning which allows us to connect this concept with those of elementary analytic geometry. The derivative at a point is the angular coefficient of the tangent line at the point, in other words we have this unifying relation:

Furthermore, the derivative also has a powerful physical meaning. In fact, every physical quantity that can be expressed through a ratio is the derivative of the function in the numerator in relation to the denominator. For example, velocity is the first derivative of space with respect to time, acceleration the first derivative of velocity with respect to time, and so on for all the almost infinite physical quantities definable in this way.

IV

Impressum

Verlag: BookRix GmbH & Co. KG

Tag der Veröffentlichung: 19.04.2023
ISBN: 978-3-7554-3937-0

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