Let us make a number of explanatory remarks about the specification of a function by an analytic expression or formula, which play an extremely important role in mathematical analysis.

1° First of all, what analytical operations or actions can be included in these formulas? In the first place here are understood all the operations studied in elementary algebra and trigonometry: arithmetic operations, exponentiation (and root extraction), logarithm, transition from angles to their trigonometric values ​​and vice versa [see. below 48 - 51]. However, and it is important to emphasize this, as our information on analysis develops, other operations will be added to their number, first of all, the passage to the limit, with which the reader is already familiar from Chapter I.

Thus, the full content of the term "analytical expression" or "formula" will be revealed only gradually.

2° The second remark relates to the domain of definition of a function by an analytic expression or formula.

Each analytic expression containing an argument x has, so to speak, a natural area of ​​application: it is the set of all those values ​​of x for which it retains a meaning, i.e., has a well-defined, finite, real value. Let's explain this with simple examples.

So, for an expression, such an area will be the whole set of real numbers. For an expression, this area will be reduced to a closed interval beyond which its value ceases to be real. On the contrary, the expression will have to include an open gap as its natural scope, because at the ends its denominator becomes 0. Sometimes the range of values ​​for which the expression retains meaning consists of scattered gaps: for these there will be gaps for - gaps, etc.

As a final example, consider the sum of an infinite geometric progression

If then, as we know, this limit exists and has a value of . For , the limit is either equal or does not exist at all. Thus, for the above analytic expression, the natural scope will be the open interval

In the following presentation, we will have to consider both more complex and more general analytic expressions, and we will more than once study the properties of functions given by such an expression in the entire region where it retains meaning, i.e., the study of the analytic apparatus itself.

However, another state of affairs is also possible, to which we consider it necessary to draw the reader's attention in advance. Let us imagine that some particular question, in which the variable x is essentially limited to the range of X, led to the consideration of a function admitting an analytic expression. Although it may happen that this expression makes sense outside the region X, it is, of course, impossible to go beyond it. Here the analytical expression plays a subordinate, auxiliary role.

For example, if, investigating the free fall of a heavy point from a height above the earth's surface, we resort to the formula

It would be absurd to consider negative values ​​of t or values ​​greater than for, as it is easy to see, at , the point will already fall to the ground. And this is despite the fact that the expression itself - retains its meaning for all real .

3° It may happen that a function is not defined by the same formula for all values ​​of the argument, but for some by one formula and for others by another. An example of such a function in between is the function defined by the following three formulas:

and finally if .

We also mention the Dirichlet function (P. G. Lejeune-Dinchlet), which is defined as follows:

Finally, together with Kronecker (L. Kroneckcf) we will consider the function, which he called "signum" and denoted by

Various ways of setting a function Analytical, graphical, tabular - the simplest, and therefore the most popular ways of setting a function, for our needs these methods are quite enough. Analytical graphic tabular In fact, in mathematics there are quite a few different ways of specifying a function, and one of them is verbal, which is used in very peculiar situations.

Verbal way of specifying a function A function can also be specified verbally, that is, descriptively. For example, the so-called Dirichlet function is defined as follows: the function y is equal to 0 for all rational and 1 for all irrational values ​​of the argument x. Such a function cannot be defined by a table, since it is defined on the entire number axis and the set of values ​​for its argument is infinite. Graphically, this function cannot be defined either. Nevertheless, an analytical expression for this function was found, but it is so complicated that it has no practical value. The verbal method gives a short and clear definition of it.

Example 1 The function y = f (x) is defined on the set of all non-negative numbers using the following rule: each number x 0 is assigned the first decimal place in the decimal representation of the number x. If, say, x \u003d 2.534, then f (x) \u003d 5 (the first decimal place is the number 5); if x = 13.002, then f(x) = 0; if x \u003d 2/3, then, writing 2/3 as an infinite decimal fraction 0.6666 ..., we find f (x) \u003d 6. And what is the value of f (15)? It is equal to 0, since 15 = 15.000…, and we see that the first decimal place after the decimal point is 0 (actually, the equality 15 = 14.999… is true, but mathematicians agreed not to consider infinite periodic decimal fractions with a period of 9).

Any non-negative number x can be written as a decimal fraction (finite or infinite), and therefore for each value of x you can find a certain number of values ​​\u200b\u200bof the first decimal place, so we can talk about a function, albeit somewhat unusual. D (f) = . = 2 [" title="(!LANG: A function that is defined by the conditions: f (x) is an integer; f (x) x; x; f + 1 > x,x, the integer part of the number is called the integer part of the number. D (f) = (-;+), E (f) = Z (set of integers) For the integer part of the number x, use the notation [ x ].= 2 [" class="link_thumb"> 7 !} A function that is determined by the conditions: f (x) is an integer; f(x)x;x; f + 1 > x,x, the integer part of the number is called the integer part of the number. D (f) \u003d (-;+), E (f) \u003d Z (set of integers) For the integer part of the number x, the notation [ x ] is used. = 2 = 47 [-0.23] = - 1 x,x, the integer part of the number is called the integer part of the number. D (f) \u003d (-;+), E (f) \u003d Z (set of integers) For the integer part of the number x, the notation [ x ] is used. \u003d 2 ["\u003e x, x, the integer part of the number is called the integer part of the number. D (f) \u003d (-; +), E (f) \u003d Z (set of integers) For the integer part of the number x, the notation [x] is used. \u003d 2 \u003d 47 [ - 0.23] \u003d - 1 "\u003e x, x, the integer part of the number is called the integer part of the number. D (f) \u003d (-;+), E (f) \u003d Z (set of integers) For the integer part of the number x, the notation [ x ] is used. = 2 [" title="(!LANG: A function that is defined by the conditions: f (x) is an integer; f (x) x; x; f + 1 > x,x, the integer part of the number is called the integer part of the number. D (f) = (-;+), E (f) = Z (set of integers) For the integer part of the number x, use the notation [ x ].= 2 ["> title="A function that is determined by the conditions: f (x) is an integer; f(x)x;x; f + 1 > x,x, the integer part of the number is called the integer part of the number. D (f) \u003d (-;+), E (f) \u003d Z (set of integers) For the integer part of the number x, the notation [ x ] is used. = 2["> !}

Of all the above methods of specifying a function, the analytical method provides the greatest opportunities for using the apparatus of mathematical analysis, and the graphic method has the greatest clarity. That is why mathematical analysis is based on a deep synthesis of analytical and geometric methods. The study of functions given analytically is much easier and becomes clear if we consider the graphs of these functions in parallel.

X y=x

Great mathematician - Dirichlet In professor at Berlin, from 1855 Göttingen University. The main works on number theory and mathematical analysis. In the field of mathematical analysis, Dirichlet for the first time accurately formulated and investigated the concept of conditional convergence of a series, established a criterion for the convergence of a series (the so-called Dirichlet criterion, 1862), gave (1829) a rigorous proof of the possibility of expanding a function into a Fourier series having a finite number of maxima and minima. Significant works of Dirichlet are devoted to mechanics and mathematical physics (Dirichlet's principle in the theory of harmonic function). Dirichlet Peter Gustav Lejeune () German mathematician, foreign corresponding member. Petersburg Academy of Sciences (c), member of the Royal Society of London (1855), Parisian Academy of Sciences (1854), Berlin Academy of Sciences. Dirichlet proved a theorem on the existence of an infinitely large number of primes in any arithmetic progression of integers, the first term and the difference of which are coprime numbers and studied (1837) the law of distribution of primes in arithmetic progressions, in connection with which he introduced functional series of a special form ( so-called Dirichlet series).

One of the classic definitions of the concept of "function" are definitions based on correspondences. We present a number of such definitions.

Definition 1

A relationship in which each value of the independent variable corresponds to a single value of the dependent variable is called function.

Definition 2

Let two non-empty sets $X$ and $Y$ be given. A match $f$ that maps to each $x\in X$ one and only one $y\in Y$ is called function($f:X → Y$).

Definition 3

Let $M$ and $N$ be two arbitrary numerical sets. It is said that a function $f$ is defined on $M$, taking values ​​from $N$ if each element of $x\in X$ is associated with one and only one element from $N$.

The following definition is given through the concept of a variable. A variable is a quantity that in this study takes on various numerical values.

Definition 4

Let $M$ be the set of values ​​of the variable $x$. Then, if each value $x\in M$ corresponds to one definite value of another variable $y$ is a function of the value $x$ defined on the set $M$.

Definition 5

Let $X$ and $Y$ be some number sets. A function is a set $f$ of ordered pairs of numbers $(x,\ y)$ such that $x\in X$, $y\in Y$ and each $x$ belongs to one and only one pair of this set, and each $y$ is in at least one pair of .

Definition 6

Any set $f=\(\left(x,\ y\right)\)$ of ordered pairs $\left(x,\ y\right)$ such that for any pairs $\left(x",\ y" \right)\in f$ and $\left(x"",\ y""\right)\in f$ it follows from the condition $y"≠ y""$ that $x"≠x""$ is called a function or display.

Definition 7

A function $f:X → Y$ is a set $f$ of ordered pairs $\left(x,\ y\right)\in X\times Y$ such that for any element $x\in X$ there is a unique element $y\in Y$ such that $\left(x,\ y\right)\in f$, that is, the function is a tuple of objects $\left(f,\ X,\ Y\right)$.

In these definitions

$x$ is an independent variable.

$y$ is the dependent variable.

All possible values ​​of the variable $x$ are called the domain of the function, and all possible values ​​of the variable $y$ are called the domain of the function.

Analytical way of defining a function

For this method, we need the concept of an analytic expression.

Definition 8

An analytic expression is the product of all possible mathematical operations on any numbers and variables.

The analytical way of setting a function is its setting using an analytical expression.

Example 1

$y=x^2+7x-3$, $y=\frac(x+5)(x+2)$, $y=cos5x$.

Pros:

1. With formulas, we can determine the value of a function for any given value of the variable $x$;
2. Functions defined in this way can be studied using the apparatus of mathematical analysis.

Minuses:

1. Little visibility.
2. Sometimes you have to perform very cumbersome calculations.

Tabular way of defining a function

This way of setting is that for several values ​​of the independent variable, the values ​​of the dependent variable are written out. All this is entered into the table.

Example 2

Picture 1.

A plus: For any value of the independent variable $x$ that is entered in the table, the corresponding value of the function $y$ is immediately recognized.

Minuses:

1. More often than not, there is no full specification of the function;
2. Little visibility.

a function is a correspondence between elements of two sets, established according to such a rule that each element of one set is associated with some element from another set.

the graph of a function is the locus of points in the plane whose abscissas (x) and ordinates (y) are connected by the specified function:

the point is located (or is located) on the graph of the function if and only if .

Thus, a function can be adequately described by its graph.

tabular way. Quite common, it consists in setting a table of individual argument values ​​​​and their corresponding function values. This method of defining a function is used when the domain of the function is a discrete finite set.

With the tabular method of specifying a function, it is possible to approximately calculate the values ​​of the function that are not contained in the table, corresponding to the intermediate values ​​of the argument. To do this, use the method of interpolation.

The advantages of the tabular method of setting a function are that it makes it possible to determine certain specific values ​​​​at once, without additional measurements or calculations. However, in some cases, the table does not define the function completely, but only for some values ​​of the argument and does not provide a visual representation of the nature of the change in the function depending on the change in the argument.

Graphic way. The graph of the function y = f(x) is the set of all points in the plane whose coordinates satisfy the given equation.

The graphical way of specifying a function does not always make it possible to accurately determine the numerical values ​​of the argument. However, it has a great advantage over other methods - visibility. In engineering and physics, a graphical method of setting a function is often used, and a graph is the only way available for this.

In order for the graphical assignment of a function to be quite correct from a mathematical point of view, it is necessary to indicate the exact geometric construction of the graph, which, most often, is given by an equation. This leads to the following way of defining a function.

analytical way. Most often, the law that establishes a relationship between an argument and a function is specified by means of formulas. This way of defining a function is called analytical.

This method makes it possible for each numerical value of the argument x to find the corresponding numerical value of the function y exactly or with some accuracy.

If the relationship between x and y is given by a formula that is resolved with respect to y, i.e. has the form y = f(x), then we say that the function of x is given explicitly.

If the values ​​x and y are related by some equation of the form F(x,y) = 0, i.e. the formula is not allowed with respect to y, which means that the function y = f(x) is implicitly defined.

A function can be defined by different formulas in different parts of its task area.

The analytical method is the most common way to define functions. Compactness, conciseness, the ability to calculate the value of a function for an arbitrary value of the argument from the domain of definition, the ability to apply the apparatus of mathematical analysis to a given function are the main advantages of the analytical method of defining a function. The disadvantages include the lack of visibility, which is compensated by the ability to build a graph and the need to perform sometimes very cumbersome calculations.

verbal way. This method consists in the fact that the functional dependence is expressed in words.

Example 1: the function E(x) is the integer part of the number x. In general, E(x) = [x] denotes the largest integer that does not exceed x. In other words, if x = r + q, where r is an integer (may be negative) and q belongs to the interval = r. The function E(x) = [x] is constant on the interval = r.

Example 2: function y = (x) - fractional part of a number. More precisely, y =(x) = x - [x], where [x] is the integer part of the number x. This function is defined for all x. If x is an arbitrary number, then representing it as x = r + q (r = [x]), where r is an integer and q lies in the interval .
We see that adding n to the x argument does not change the value of the function.
The smallest non-zero number in n is , thus the period is sin 2x .

The value of the argument for which the function is equal to 0 is called zero (root) functions.

A function can have multiple zeros.

For example, the function y=x(x+1)(x-3) has three zeros: x=0, x=-1, x=3.

Geometrically, the zero of a function is the abscissa of the intersection point of the graph of the function with the axis X .

Figure 7 shows the graph of the function with zeros: x = a, x = b and x = c .

If the graph of a function approaches a certain straight line indefinitely as it moves away from the origin, then this straight line is called asymptote.

Inverse function

Let the function y=ƒ(x) be given with the domain of definition D and the set of values ​​E. If each value yєE corresponds to a single value xєD, then the function x=φ(y) is defined with the domain of definition E and the set of values ​​D (see Fig. 102 ).

Such a function φ(y) is called the inverse of the function ƒ(x) and is written in the following form: x=j(y)=f -1 (y). About the functions y=ƒ(x) and x=φ(y) they say that they are mutually inverse. To find the function x=φ(y) inverse to the function y=ƒ(x), it is sufficient to solve the equation ƒ(x)=y with respect to x (if possible).

1. For the function y \u003d 2x, the inverse function is the function x \u003d y / 2;

2. For the function y \u003d x2 xє, the inverse function is x \u003d √y; note that for the function y \u003d x 2, given on the segment [-1; 1], there is no inverse, since one value of y corresponds to two values ​​of x (for example, if y=1/4, then x1=1/2, x2=-1/2).

It follows from the definition of the inverse function that the function y=ƒ(x) has an inverse if and only if the function ƒ(x) defines a one-to-one correspondence between the sets D and E. It follows that any strictly monotonic function has an inverse. Moreover, if the function increases (decreases), then the inverse function also increases (decreases).

Note that the function y \u003d ƒ (x) and its inverse x \u003d φ (y) are depicted by the same curve, that is, their graphs coincide. If we agree that, as usual, the independent variable (i.e., the argument) is denoted by x, and the dependent variable by y, then the inverse function of the function y \u003d ƒ (x) will be written as y \u003d φ (x).

This means that the point M 1 (x o; y o) of the curve y=ƒ(x) becomes the point M 2 (y o; x o) of the curve y=φ(x). But the points M 1 and M 2 are symmetrical about the straight line y \u003d x (see Fig. 103). Therefore, the graphs of mutually inverse functions y=ƒ(x) and y=φ(x) are symmetrical with respect to the bisector of the first and third coordinate angles.

Complex function

Let the function y=ƒ(u) be defined on the set D, and the function u= φ(x) on the set D 1 , and for  x D 1 the corresponding value u=φ(x) є D. Then on the set D 1 is defined function u=ƒ(φ(x)), which is called a complex function of x (or a superposition of given functions, or a function of a function).

The variable u=φ(x) is called an intermediate argument of a complex function.

For example, the function y=sin2x is a superposition of two functions y=sinu and u=2x. A complex function can have multiple intermediate arguments.

4. Basic elementary functions and their graphs.

The following functions are called basic elementary functions.

1) The exponential function y \u003d a x, a> 0, a ≠ 1. In fig. 104 shows graphs of exponential functions corresponding to various exponential bases.

2) Power function y=x α , αєR. Examples of graphs of power functions corresponding to various exponents are provided in the figures

3) Logarithmic function y=log a x, a>0,a≠1; Graphs of logarithmic functions corresponding to different bases are shown in fig. 106.

4) Trigonometric functions y=sinx, y=cosx, y=tgx, y=ctgx; Graphs of trigonometric functions have the form shown in fig. 107.

5) Inverse trigonometric functions y=arcsinx, y=arccosx, y=arctgx, y=arcctgx. On fig. 108 shows graphs of inverse trigonometric functions.

A function given by one formula, composed of basic elementary functions and constants using a finite number of arithmetic operations (addition, subtraction, multiplication, division) and operations of taking a function from a function, is called an elementary function.

Examples of elementary functions are the functions

Examples of non-elementary functions are the functions

5. Concepts of the limit of a sequence and a function. Limit properties.

Function limit (function limit) at a given point, limiting for the domain of definition of a function, is such a value to which the value of the function under consideration tends when its argument tends to a given point.

In mathematics sequence limit elements of a metric space or a topological space is an element of the same space that has the property of "attracting" elements of a given sequence. The limit of a sequence of elements of a topological space is such a point, each neighborhood of which contains all the elements of the sequence, starting from some number. In a metric space, neighborhoods are defined in terms of a distance function, so the concept of a limit is formulated in the language of distances. Historically, the first was the concept of the limit of a numerical sequence, which arises in mathematical analysis, where it serves as the basis for a system of approximations and is widely used in the construction of differential and integral calculus.

Designation:

(read: the limit of the x-nth sequence as en tending to infinity is a)

The property of a sequence to have a limit is called convergence: if a sequence has a limit, then the given sequence is said to be converges; otherwise (if the sequence has no limit) the sequence is said to be diverges. In a Hausdorff space, and in particular a metric space, every subsequence of a convergent sequence converges, and its limit is the same as the limit of the original sequence. In other words, a sequence of elements in a Hausdorff space cannot have two different limits. It may, however, turn out that the sequence has no limit, but there is a subsequence (of the given sequence) that has a limit. If a convergent subsequence can be distinguished from any sequence of points in a space, then the space is said to have the property of sequential compactness (or, simply, compactness if compactness is defined exclusively in terms of sequences).

The concept of the limit of a sequence is directly related to the concept of a limit point (set): if a set has a limit point, then there is a sequence of elements of the given set converging to the given point.

Definition

Let a topological space and a sequence be given Then, if there exists an element such that

where is an open set containing , then it is called the limit of the sequence . If the space is metric, then the limit can be defined using a metric: if there exists an element such that

where is the metric, then is called the limit.

· If a space is equipped with an antidiscrete topology, then the limit of any sequence is any element of the space.

6. Limit of a function at a point. Unilateral limits.

Function of one variable. Determining the limit of a function at a point according to Cauchy. Number b is called the limit of the function at = f(x) at X striving for a(or at the point a) if for any positive number  there is a positive number  such that for all x ≠ a, such that | x – a | < , выполняется неравенство
| f(x) – a | <  .

Determining the limit of a function at a point according to Heine. Number b is called the limit of the function at = f(x) at X striving for a(or at the point a) if for any sequence ( x n ) converging to a(aspiring to a, which has a limit number a), and for any value n x n≠ a, subsequence ( y n= f(x n)) converges to b.

These definitions assume that the function at = f(x) is defined in some neighborhood of the point a, except perhaps for the very point a.

The definitions of the limit of a function at a point according to Cauchy and according to Heine are equivalent: if the number b serves as a limit in one of them, then the same is true in the second.

The specified limit is indicated as follows:

Geometrically, the existence of a function limit at a point according to Cauchy means that for any number  > 0, one can indicate such a rectangle on the coordinate plane with a base 2 > 0, a height 2 and a center at the point ( a; b) that all points of the graph of this function on the interval ( a– ; a+ ), with the possible exception of the point M(a; f(a)), lie in this rectangle

One-sided limit in mathematical analysis, the limit of a numerical function, implying "approaching" the limit point from one side. Such limits are called respectively left-hand limit(or left limit) and right-hand limit (limit on the right). Let a numerical function be given on some numerical set and the number be the limit point of the domain of definition. There are various definitions for the one-sided limits of a function at a point, but they are all equivalent.

is given, in other words, known, if for each value of the possible number of arguments it is possible to find out the corresponding value of the function. The most common three function definition method: tabular, graphic, analytical, there are also verbal and recursive methods.

1. Tabular way the most widespread (tables of logarithms, square roots), its main advantage is the possibility of obtaining a numerical value of the function, the disadvantages are that the table can be difficult to read and sometimes does not contain intermediate values ​​of the argument.

For example:

x
y

Argument X takes the values ​​specified in the table, and at defined according to this argument X.

2. Graphical way consists in drawing a line (graph), in which the abscissas represent the values ​​of the argument, and the ordinates represent the corresponding values ​​of the function. Often, for clarity, the scales on the axes are taken different.

For example: to find the schedule at, which corresponds to x = 2.5 it is necessary to draw a perpendicular to the axis X at the mark 2,5 . The mark can be quite accurately done with a ruler. Then we find that at X = 2,5 at equals 7,5 , but if we need to find the value at at X equal to 2,76 , then the graphical way of setting the function will not be accurate enough, because The ruler does not allow for such an accurate measurement.

The advantages of this method of setting functions are in the ease and integrity of perception, in the continuity of the change of the argument; the disadvantage is a decrease in the degree of accuracy and the difficulty of obtaining accurate values.

3. Analytical method consists in specifying a function by one or more formulas. The main advantage of this method is the high accuracy of determining the function of the argument of interest, and the disadvantage is the time spent on additional mathematical operations.

For example:

The function can be specified using the mathematical formula y=x2, then if X equals 2 , then at equals 4, we are building X into a square.

4. verbal way consists in defining the function in plain language, i.e. words. In this case, it is necessary to give input, output values ​​and the correspondence between them.

For example:

You can verbally specify a function (task) that is accepted as a natural argument X with the corresponding value of the sum of the digits that make up the value at. Explain: if X equals 4 , then at equals 4 , what if X equals 358 , then at is equal to the sum 3 + 5 + 8 , i.e. 16 . Further similarly.

5. Recursive way consists in specifying a function through itself, while function values are defined in terms of its other values. This way of defining a function is used in defining sets and series.

For example:

When decomposed Euler numbers given by the function:

Its abbreviation is given below:

In direct calculation, infinite recursion occurs, but it can be proved that the value f(n) with increasing n tends to unity (therefore, despite the infinity of the series , the value Euler numbers certainly). For an approximate calculation of the value e it is enough to artificially limit the recursion depth to some predetermined number and, upon reaching it, use it instead f(n) unit.