The concept of a function Ways of defining a function Examples of functions Analytical definition of a function Graphical way of defining a function Limit of a function at a point Tabular way of defining a function Limit theorems Uniqueness of a limit Boundedness of a function that has a limit Passing to a limit at inequality Limit of a function at infinity Infinitesimal functions Properties of infinitesimal functions

The concept of a function is basic and original, as is the concept of a set. Let X be some set of real numbers x. If a certain real number y is assigned to each x ∈ X according to some law, then they say that a function is given on the set X and write. The function introduced in this way is called a numerical one. In this case, the set X is called the domain of the definition of the function, and the independent variable x is called the argument. To indicate a function, sometimes only the symbol is used, which denotes the law of correspondence, i.e. instead of f (x) n and jester, just /. Thus, the function is given if 1) the domain of definition is specified 2) the rule /, which assigns to each value a: € X a certain number y \u003d / (x) - the value of the function corresponding to this value of the argument x. The functions / and g are called equal if their domains of definition coincide and the equality f(x) = g(x) is true for any value of the argument x from their common domain. Thus, the functions y are not equal; they are equal only on the interval [O, I]. Function examples. 1. The sequence (o„) is a function of an integer argument, defined on the set of natural numbers, such that f(n) = an (n = 1,2,...). 2. Function y = n? (read "en-factorial"). Given on the set of natural numbers: each natural number n is associated with the product of all natural numbers from 1 to n inclusive: moreover, 0! = 1. The designation sign comes from the Latin word signum - a sign. This function is defined on the entire number line; the set of its values ​​consists of three numbers -1.0, I (Fig. 1). y = |x), where (x) denotes the integer part of a real number x, i.e. [x| - the largest integer not exceeding It is read: - the game is equal to antie x ”(fr. entier). This function is set on the entire number axis, and the set of all its values ​​consists of integers (Fig. 2). Ways of Specifying a Function Analytical Specifying a Function A function y = f(x) is said to be specified analytically if it is defined using a formula that indicates what operations must be performed on each value of x in order to obtain the corresponding value of y. For example, the function is given analytically. In this case, the domain of the function (if it is not specified in advance) is understood as the set of all real values ​​of the argument x, for which the analytical expression that defines the function takes only real and final values. In this sense, the domain of a function is also called its domain of existence. For the function, the domain of definition is the segment. For the function y - sin x, the domain of definition is the entire numerical axis. Note that not every formula defines a function. For example, the formula does not define any function, since there is not a single real value of x for which both roots written above would have real values. Analytical assignment of a function can look rather complicated. In particular, a function can be defined by different formulas on different parts of its domain of definition. For example, a function could be defined like this: 1.2. Graphical way of specifying a function The function y = f(x) is called specified graphically if its schedule is specified, i.e. a set of points (xy/(x)) on the xOy plane, the abscissas of which belong to the domain of definition of the function, and the ordinates are equal to the corresponding values ​​of the function (Fig. 4). Not for every function, its graph can be depicted in the figure. For example, the Dirichlet function if x is rational, if x is irrational, ZX \o, does not allow such a representation. The function R(x) is given on the entire numerical axis, and the set of its values ​​consists of two numbers 0 and 1. 1.3. Tabular way of specifying a function A function is said to be specified tabular if a table is provided that contains the numerical values ​​of the function for some values ​​of the argument. When a function is defined in a table, its domain of definition consists only of the values ​​x\t x2i..., xn listed in the table. §2. Limit of a function at a point The concept of the limit of a function is central to mathematical analysis. Let the function f(x) be defined in some neighborhood Q of the point xq, except, perhaps, for the extension (Cauchy) point itself. The number A is called the limit of the function f(x) at the point x0 if for any number e > 0, which can be arbitrarily small, there exists a number<5 > 0, such that for all iGH.i^ x0 satisfying the condition the inequality is true Concept of a function Ways of defining a function Examples of functions Analytical definition of a function Graphical way of defining a function Limit of a function at a point Tabular way of defining a function limit theorem uniqueness of a limit boundedness of a function that has a limit transition to the limit in the inequality Limit of a function at infinity Infinitesimal functions Properties of infinitesimal functions Notation: With the help of logical symbols, this definition is expressed as follows. Examples. 1. Using the definition of the limit of a function at a point, show that the Function is defined everywhere, including the point zo = 1: /(1) = 5. Take any. In order for the inequality |(2x + 3) - 5| took place, it is necessary to fulfill the following inequalities Therefore, if we take we will have. This means that the number 5 is the limit of the function: at point 2. Using the definition of the limit of a function, show that the function is not defined at the point xo = 2. Consider /(x) in some neighborhood of the point-Xq = 2, for example, on the interval ( 1, 5) that does not contain the point x = 0, at which the function /(x) is also not defined. Take an arbitrary number c > 0 and transform the expression |/(x) - 2| for x f 2 as follows For x b (1, 5) we get the inequality From this it is clear that if we take 6 \u003d c, then for all x € (1.5) subject to the condition the inequality will be true This means that the number A - 2 is the limit of a given function at a point Let us give a geometric explanation of the concept of the limit of a function at a point, referring to its graph (Fig. 5). For x, the values ​​of the function /(x) are determined by the ordinates of the points of the curve M \ M, for x > ho - by the ordinates of the points of the curve MM2. The value /(x0) is determined by the ordinate of the point N. The graph of this function is obtained if we take the "good" curve M\MMg and replace the point M(x0, A) on the curve with the point jV. Let us show that at the point x0 the function /(x) has a limit equal to the number A (the ordinate of the point M). Take any (arbitrarily small) number e > 0. Mark on the Oy axis points with ordinates A, A - e, A + e. Denote by P and Q the points of intersection of the graph of the function y \u003d / (x) with the lines y \u003d A - enu = A + e. Let the abscissas of these points be x0 - hx0 + hi, respectively (ht > 0, /12 > 0). It can be seen from the figure that for any x Φ x0 from the interval (x0 - h\, x0 + hi) the value of the function f(x) is between. for all x ⩽ x0 satisfying the condition, the inequality is true We set Then the interval will be contained in the interval and, therefore, the inequality or, which will also be satisfied for all x satisfying the condition This proves that Thus, the function y \u003d f (x) has a limit A at the point x0 if, no matter how narrow the e-strip between the lines y = A - eny = A + e, there is such "5 > 0, such that for all x from the punctured neighborhood of the point x0 of the point of the graph of the function y = / (x) are inside the indicated e-band. Remark 1. The quantity b depends on e: 6 = 6(e). Remark 2. In the definition of the limit of a function at the point Xq, the point x0 itself is excluded from consideration. Thus, the value of the function at the Ho ns point does not affect the limit of the function at that point. Moreover, the function may not even be defined at the point Xq. Therefore, two functions that are equal in a neighborhood of the point Xq, excluding, perhaps, the point x0 itself (they may have different values ​​at it, one of them or both together may not be defined), have the same limit for x - Xq, or both have no limit. From this, in particular, it follows that in order to find the limit of a fraction at the point xo, it is legitimate to reduce this fraction by equal expressions that vanish at x = Xq. Example 1. Find The function /(x) = j for all x Ф 0 is equal to one, and at the point x = 0 it is not defined. Replacing f(x) with the function g(x) = 1 equal to it at x 0, we obtain the concept of a function Ways of defining a function Examples of functions Analytical definition of a function Graphical way of defining a function Limit of a function at a point Tabular way of defining a function Limit theorems Uniqueness of a limit Boundedness of a function having a limit transition to the limit in the inequality Limit of a function at infinity Infinitely small functions Properties of infinitely small functions x = 0 limit equal to zero: lim q(x) = 0 (show it!). Therefore, lim /(x) = 0. Problem. Formulate with the help of inequalities (in the language of e -6), which means Let the function /(n) be defined in some neighborhood Π of the point x0, except, perhaps, the point x0 itself. Definition (Heine). The number A is called the limit of the function /(x) at the point x0, if for any sequence (xn) of values ​​of the argument x 6 P, zn / x0) converging to the point x0, the corresponding sequence of values ​​of the function (/(xn)) converges to number A. It is convenient to use the above definition when it is necessary to establish that the function /(x) has no limit at the point x0. To do this, it suffices to find some sequence (/(xn)) that does not have a limit, or to indicate two sequences (/(xn)) and (/(x "n)) that have different limits. Let us show, for example, that the function iiya / (x) = sin j (Fig. 7), defined EVERYWHERE, except for the POINT X = O, Fig. 7 does not have a limit at the point x = 0. Consider two sequences (, converging to the point x = 0. The corresponding sequences values ​​of the function /(x) converge to different limits: the sequence (sinnTr) converges to zero, and the sequence (sin(5 +) converges to one. This means that the function f(x) = sin j at the point x = 0 has no limit. Comment. Both definitions of the limit of a function at a point (Cauchy's definition and Heine's definition) are equivalent. §3. Theorems on limits Theorem 1 (uniqueness of the limit). If the function f(x) has a limit at xo, then this limit is unique. A Let lim f(x) = A. Let us show that no number B φ A can be the limit x-x0 of the function f(x) at the point x0. The fact that lim / (x) φ With the help of logical symbols XO is formulated as follows: Using the inequality we obtain, Take e = > 0. Since lim / (x) = A, for the chosen e > 0 there is 6 > 0 such that From relation (1) for the indicated values ​​of x we ​​have So, it has been found that, no matter how small, there are x Φ xQ, such that and at the same time ^ e. Hence the Definition. A function /(x) is said to be bounded in a neighborhood of the point x0 if there are numbers M > 0 and 6 > 0 such that Theorem 2 (boundedness of a function that has a limit). If the function f(x) is defined in a neighborhood of the point x0 and has a finite limit at the point x0, then it is bounded in some neighborhood of this point. m Let Then for any example, for e = 1, there is such 6 > 0 that for all x φ x0 satisfying the condition, the inequality will be true Noting that we always get Let. Then at each point x of the interval we have This means, according to the definition, that the function f(x) is bounded in a neighborhood. For example, the function /(x) = sin is bounded in a neighborhood of the point but has no limit at the point x = 0. Let us formulate two more theorems, the geometric meaning of which is quite clear. Theorem 3 (passing to the limit in inequality). If /(x) ⩽ ip(x) for all x in some neighborhood of the point x0, except perhaps for the point x0 itself, and each of the functions /(x) and ip(x) at the point x0 has a limit, then Note that that a strict inequality for functions does not necessarily imply a strict inequality for their limits. If these limits exist, then we can only assert that Thus, for example, the inequality while is true for functions. Theorem 4 (limit of an intermediate function). If for all x in some neighborhood of the point Xq, except, perhaps, the point x0 itself (Fig. 9), and the functions f(x) and ip(x) at the point xo have the same limit A, then the function f (x) at the point x0 has a limit equal to the same value of A. § ​​4. Limit of a function at infinity Let the function /(x) be defined either on the entire real axis or at least for all x satisfying the condition jx| > K for some K > 0. Definition. The number A is called the limit of the function f(x) as x tends to infinity, and they write if for any e > 0 there exists a number jV > 0 such that for all x satisfying the condition |x| > X, the inequality is true Replacing the condition in this definition accordingly, we obtain definitions From these definitions it follows that if and only if simultaneously That fact, geometrically means the following: no matter how narrow the e-strip between the lines y \u003d A- euy \u003d A + e, there is such a straight line x = N > 0 that to the right carried the graph of the function y = /(x) is entirely contained in the indicated e-strip (Fig. 10). In this case, they say that for x + oo, the graph of the function y \u003d / (x) asymptotically approaches the straight line y \u003d A. Example, The function / (x) \u003d jtjj- is defined on the entire real axis and is a fraction whose numerator is constant , and the denominator increases indefinitely as |x| +oo. It is natural to expect that lim /(x)=0. Let's show it. М Take any e > 0, subject to the condition For the relation to take place, the inequality c or must be satisfied, which is the same as whence Thus. if we take we will have. This means that the number is the limit of this function at Note that the radical expression is only for t ^ 1. In the case when, the inequality c is automatically satisfied for all. The graph of an even function y = - asymptotically approaches the straight line Formulate using inequalities, which means §5. Infinitely Small Functions Let the function a(x) be defined in some neighborhood of the point x0, except possibly for the point x0 itself. Definition. The function a(x) is called an infinitesimal function (abbreviated as b.m.f.) as x tends to x0 if within the uniqueness of the limit boundedness of a function that has a limit transition to the limit in the inequality The limit of a function at infinity Infinitesimal functions Properties of infinitesimal functions For example, the function a(x) = x - 1 is b. m. f. at x 1, since lim (x-l) \u003d 0. The graph of the function y \u003d x-1 1-1 is shown in fig. II. In general, the function a(x)=x-x0 is the simplest example of b. m. f. at x-»ho. Taking into account the definition of the limit of a function at a point, the definition of b. m. f. can be formulated like this. Definition. A function a(x) is said to be infinitely small for x - * xo if for any t > 0 there exists such "5 > 0 such that for all x satisfying the condition the inequality is true functions at Definition. The function a(x) is called infinitely small for x -» oo, if then the function a(x) is called infinitesimal, respectively, for or for For example, the function is infinitesimal for x -» oo, since lim j = 0. The function a(x ) = e~x is an infinitely small function as x -* + oo, since in what follows we will, as a rule, consider all the concepts and theorems related to the limits of functions only in relation to the case of the limit of a function at a point, leaving the reader to formulate the corresponding concepts for himself and prove similar theorems of the day cases when Properties of infinitesimal functions Theorem 5. If a(x) and P(x) - b. m. f. for x - * xo, then their sum a(x) + P(x) is also a b.m. f. at x -» ho. 4 Take any e > 0. Since a(x) is a b.m.f. for x -* xo, then there is "51 > 0 such that for all x Φ xo satisfying the condition the inequality is true. By condition P(x) also b.m.f. for x ho, so there is such that for all χ φ ho satisfying the condition, the inequality is true Let us set 6 = min(«5j, 62). Then for all x Ф ho satisfying the condition, inequalities (1) and (2) will be simultaneously true. Therefore This means that the sum a(x) +/3(x) is a b.m.f. for xxq. Comment. The theorem remains valid for the sum of any finite number of functions, b. m. at x zo. Theorem 6 (product of a b.m.f. by a bounded function). If the function a(x) is b. m. f. for x -* x0, and the function f(x) is bounded in a neighborhood of the point Xo, then the product a(x)/(x) is 6. m. f. for x -» x0. By assumption, the function f(x) is bounded in a neighborhood of the point x0. This means that there are numbers 0 and M > 0 such that Let us take any e > 0. Since, by the condition, there is 62 > 0 such that for all x φ x0 satisfying the condition |x - xol, the inequality will be true Let us set i of all x f x0 satisfying the condition |x - x0|, the inequalities will be simultaneously true Therefore This means that the product a(x)/(x) is b. m.f. with Example. The function y \u003d xsin - (Fig. 12) can be considered as the product of the functions a (ar) \u003d x and f (x) \u003d sin j. The function a(a) is b. m. f. for x - 0, and the function f denote the largest of the integers 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

Now everything is as it should be. The triple is not included in the answer, because the original inequality is strict. And the six turns on, because and the function at six exists, and the inequality condition is satisfied. We have successfully solved an inequality that (in its usual form) does not exist...

This is how some knowledge and elementary logic save in non-standard cases.)

Analytical definition of a function

Function %%y = f(x), x \in X%% given in an explicit analytical way, if a formula is given that indicates the sequence of mathematical operations that must be performed with the argument %%x%% to get the value %%f(x)%% of this function.

Example

• %% y = 2 x^2 + 3x + 5, x \in \mathbb(R)%%;
• %% y = \frac(1)(x - 5), x \neq 5%%;
• %% y = \sqrt(x), x \geq 0%%.

So, for example, in physics, with a uniformly accelerated rectilinear motion, the speed of a body is determined by the formula t%% is written as: %% s = s_0 + v_0 t + \frac(a t^2)(2) %%.

Piecewise Defined Functions

Sometimes the function under consideration can be defined by several formulas that operate in different parts of the domain of its definition, in which the function argument changes. For example: $$ y = \begin(cases) x ^ 2,~ if~x< 0, \\ \sqrt{x},~ если~x \geq 0. \end{cases} $$

Functions of this kind are sometimes called constituent or piecewise. An example of such a function is %%y = |x|%%

Function scope

If the function is specified in an explicit analytical way using a formula, but the scope of the function in the form of a set %%D%% is not specified, then by %%D%% we will always mean the set of values ​​of the argument %%x%% for which this formula makes sense . So for the function %%y = x^2%%, the domain of definition is the set %%D = \mathbb(R) = (-\infty, +\infty)%%, since the argument %%x%% can take any values ​​on number line. And for the function %%y = \frac(1)(\sqrt(1 - x^2))%%, the domain of definition will be the set of values ​​%%x%% satisfying the inequality %%1 - x^2 > 0%%, m .e. %%D = (-1, 1)%%.

Benefits of Explicit Analytic Function Definition

Note that the explicit analytical way of specifying a function is quite compact (the formula, as a rule, takes up little space), easily reproduced (the formula is easy to write down), and is most adapted to performing mathematical operations and transformations on functions.

Some of these operations - algebraic (addition, multiplication, etc.) - are well known from the school mathematics course, others (differentiation, integration) will be studied in the future. However, this method is not always clear, since the nature of the dependence of the function on the argument is not always clear, and sometimes cumbersome calculations are required to find the values ​​of the function (if necessary).

Implicit function specification

The function %%y = f(x)%% is defined in an implicit analytical way, if the relation $$F(x,y) = 0 is given, ~~~~~~~~~~(1)$$ relating the values ​​of the function %%y%% and the argument %%x%%. If given argument values, then to find the value of %%y%% corresponding to a particular value of %%x%%, it is necessary to solve the equation %%(1)%% with respect to %%y%% at that particular value of %%x%%.

Given a value of %%x%%, the equation %%(1)%% may have no solution or more than one solution. In the first case, the specified value %%x%% is not in the scope of the implicit function, and in the second case it specifies multivalued function, which has more than one value for a given argument value.

Note that if the equation %%(1)%% can be explicitly solved with respect to %%y = f(x)%%, then we obtain the same function, but already defined in an explicit analytical way. So, the equation %%x + y^5 - 1 = 0%%

and the equality %%y = \sqrt(1 - x)%% define the same function.

Parametric function definition

When the dependence of %%y%% on %%x%% is not given directly, but instead the dependences of both variables %%x%% and %%y%% on some third auxiliary variable %%t%% are given in the form

$$ \begin(cases) x = \varphi(t),\\ y = \psi(t), \end(cases) ~~~t \in T \subseteq \mathbb(R), ~~~~~ ~~~~~(2) $$they talk about parametric the method of setting the function;

then the auxiliary variable %%t%% is called a parameter.

If it is possible to exclude the parameter %%t%% from the equations %%(2)%%, then they come to a function given by an explicit or implicit analytical dependence of %%y%% on %%x%%. For example, from the relations $$ \begin(cases) x = 2 t + 5, \\ y = 4 t + 12, \end(cases), ~~~t \in \mathbb(R), $$ except for the parameter % %t%% we get the dependence %%y = 2 x + 2%%, which sets a straight line in the %%xOy%% plane.

Graphical way

An example of a graphical definition of a function

The above examples show that the analytical way of defining a function corresponds to its graphic image, which can be considered as a convenient and visual form of describing a function. Sometimes used graphic way defining a function when the dependence of %%y%% on %%x%% is given by a line on the %%xOy%% plane. However, for all its clarity, it loses in accuracy, since the values ​​of the argument and the corresponding values ​​of the function can be obtained from the graph only approximately. The resulting error depends on the scale and accuracy of measuring the abscissa and ordinate of the individual points of the graph. In the future, we will assign the function graph only the role of illustrating the behavior of the function and therefore we will restrict ourselves to constructing "sketches" of graphs that reflect the main features of functions.

Tabular way

Note tabular way function assignments, when some argument values ​​and their corresponding function values ​​are placed in a table in a certain order. This is how the well-known tables of trigonometric functions, tables of logarithms, etc. are constructed. In the form of a table, the relationship between the quantities measured in experimental studies, observations, and tests is usually presented.

The disadvantage of this method is the impossibility of directly determining the values ​​of the function for the values ​​of the argument that are not included in the table. If there is confidence that the values ​​of the argument not presented in the table belong to the domain of the considered function, then the corresponding values ​​of the function can be calculated approximately using interpolation and extrapolation.

Example

Algorithmic and verbal ways of specifying functions

The function can be set algorithmic(or programmatic) in a way that is widely used in computer calculations.

Finally, it may be noted descriptive(or verbal) a way of specifying a function, when the rule for matching the values ​​of the function to the values ​​of the argument is expressed in words.

For example, the function %%[x] = m~\forall (x \in )