Definition 1. The function is called differentiable
at the point 
, if its increment at this point can be represented as
,
(2.1)
Where 
and does not depend on 
, A 
at 
.
Theorem 1. Function 
, differentiable at the point 
if and only if it has a finite derivative at this point 
.
Proof.Necessity. Let the function 
differentiable at the point 
, i.e. equality (2.1) holds. Dividing it into 
, we get 
. Going to the limit at 
, we see that 
, i.e. the limit of the right side exists and is equal to A, which means there is also a limit on the left side, i.e. 
, and 
.
Adequacy. Let it exist 
. Then, by Theorem 1 of § 16 of Chapter 1 
, Where 
- endlessly small function at 
. Hence, i.e. the function is differentiable at the point 
.
The theorem is proven.
Comment. From Theorem 1 it follows that the concepts of a function having a finite derivative and a differentiable function are equivalent. Therefore, a function that has a finite derivative can be called differentiable, which is what the authors of some textbooks do.
How are the properties of continuity and differentiability of functions related to each other? Takes place
Theorem 2.
If the function 
differentiable at the point 
, then it is continuous at this point.
Proof. Because at the point 
, we have, which means the continuity of the function at the point 
.
The theorem is proven.
The converse is not true, that is, there are continuous functions, which are not differentiable.
Example 1. Let us show that the function 
continuous but not differentiable at a point 
.
Solution. Let's find the increment of the function at the point 
, corresponding to the increment 
argument. We have. That's why 
, that is, the function 
continuous at a point 
. On the other side,, 
, that is, one-sided derivatives at the point 
are not equal, therefore, this function at this point is not differentiable.
In mathematical analysis there are examples of functions that are continuous at each point on the number line, but not differentiable. They have a complex design.
Theorem 3. Let the function 
has at the point 
derivative 
, function 
has at the corresponding point 
derivative 
. Then the complex function 
has at the point 
derivative
or, in short, 
.
Proof. Let's give the value 
increment 
. Then we get the corresponding increment 
functions 
and increment 
functions 
. By Theorem 1 we have
, Where 
at 
.
.
Note that if 
, then 
by Theorem 2, therefore 
. Hence,.
Since there is a limit on the right side of the equality, there is also a limit on the left side and
.
The theorem is proven.
Comment. Theorem 3 has been proven for the case when the complex function 
has one intermediate variable 
. If there are several intermediate variables, then the derivative is calculated in a similar way. For example, if 
,
,
, That.
§ 3. Rules of differentiation. Derivatives of basic elementary functions
Theorem 1. Let the function 
, continuous, strictly monotonic on the segment 
and differentiable at the interior point 
this segment, and 
. Then the inverse function 
differentiable at the point 
, and 
.
Proof. Note that under the conditions of the theorem the inverse function 
exists, is continuous and strictly monotonic on the interval 
by virtue of the theorem from § 19 of Chapter 1.
Let's give it meaning 
increment 
. Then 
will receive an increment
(since the function 
strictly monotonic). Therefore we can write 
. Since when 
due to the continuity of the inverse function and 
and, by assumption, there exists 
, we have 
. This implies the existence 
and equality 
. The theorem is proven.
Example 1. Find the derivatives of the functions arcsin x,arccos x,arctg x,arcctg x/
Solution. By Theorem 1 we have (since 
, we have 
and take the root with a plus sign).
Likewise,
Theorem 2. If the functions 
And 
have derivatives at the point 
, then at the point 
have derivatives and functions 
(If 
) and the formulas are valid
A)
;b)
;V)
.
Proof.A) Let 
. Let's give 
increment 
. Then the functions u,v,y will receive increments 
, and
. From here 
and equality A) has been proven.
b) Let 
. Same as point A) we have
,
,, i.e. the formula holds b).
V) Let 
. We have 
,
,
, i.e. the formula holds V).
The theorem is proven.
Consequences. 1) If 
, That 
.
2) Formula A) holds for any finite number terms.
Proof. 1) Because 
, we have.
IN general case Corollaries 2) and 3) are proved by the method of mathematical induction.
Consider the exponential function 
, Where u
And v– some functions from X. Let's find the derivative of the function at at the point at which the functions are differentiable u
And v.To do this, imagine the function at in the form 
.By the rule of differentiation of a complex function, by virtue of Theorem 2 and Example 1 of § 1 we have
Thus,
Note that in the resulting formula the first term is the result of differentiation 
How exponential function, and the second – as power function. The differentiation technique used is called logarithmic differentiation
. It can also be convenient to use when the function being differentiated is the product of several factors.
Let us now move on to the parametric specification of functions. If function dependency at from argument X is not set directly, but using some third variable t, called parameter, formulas
,
(3.1)
then they say that the function at from X specified parametrically.
If X And at considered as rectangular coordinates of a point on the plane, then equations (3.1) are associated with each value 
point 
on a plane. With change t dot 
describes some curve on the plane. Equations (3.1) are called parametric equations of this curve. For example, the equations
(3.2)
are parametric equations of an ellipse with semi-axes A And b.
If in (3.1) the equation 
permitted relatively t,
, That parametric specification functions can be reduced to explicit:
.
Let's find the derivative 
function specified parametrically. To do this, assume that the functions 
And 
differentiable, and 
on some interval, and for the function 
there is an inverse function 
, having a finite derivative 
. Then, according to the rule of differentiation of complex and inverse functions we find: 
. Thus,
.
(3.3)
For example, derivative 
function defined by equations (3.2) has the form
.
Equation of a tangent to a curve defined parametrically at a point 
corresponding to the parameter value 
, is obtained from equation (1.4), if instead of 
substitute 
:
,
from here at 
we have
.
(3.4)
Similarly, from equation (1.5) we obtain the normal equation:
or. (3.5)
Let us now write down summary tables of derivatives of the basic elementary functions and the rules of differentiation obtained earlier.
Rules of differentiation
1.
.
2.
.
3.
.
4.
.
5. If 
, That 
. 6. If 
That 
.
7. If 
is the inverse function, then 
.
8..
Table of derivatives of basic elementary functions
1.
, Where 
.
2.
, in particular, 
3.
.
4.
.
.
5.
.
6.
.
7.
.
8.
.
9.
.
10.
.
11.
, in particular, 
.
12.
, in particular, 
.
“Is the statement S true?” is perhaps the most typical question in mathematics, when the statement is of the form: “Every element of class A also belongs to class B: A B.” To prove that such a statement is true means to prove the inclusion of A in B. To prove that it is false means to find an element of class A that does not belong to class B, in other words, to give a counterexample. For example, if the statement S is: “Every continuous function is differentiable at some point,” then the sets A and B consist respectively of all continuous functions and all functions differentiable at some points. Weierstrass’s famous example of a continuous but nowhere differentiable function f is counterexample to the inclusion of A in B, since f is an element of A that does not belong to B. At the risk of falling into oversimplification, we can say that mathematics (except for definitions, statements and calculations) consists of two parts - proofs and counterexamples, and mathematical discoveries consist of finding evidence and constructing counterexamples.
This determines the relevance of counterexamples during the formation and development of mathematics.
Most math books is devoted to proving true statements.
Generally speaking, examples in mathematics are of two types - illustrative examples and counterexamples. The former show why this or that statement makes sense, and the latter show why this or that statement is meaningless. It can be argued that any example is at the same time a counterexample to some statement, namely to the statement that such an example is impossible. We do not wish to give such a universal meaning to the term counterexample, but we admit that its meaning is broad enough to include all examples whose role is not limited to illustrating true theorems. So, for example, a polynomial as an example of a continuous function is not a counterexample, but a polynomial as an example of an unbounded or non-periodic function is a counterexample. In the same way, the class of all monotone functions on a bounded closed interval as a class of integrable functions is not a counterexample, but this same class as an example of a functional, but not a vector, space is a counterexample.
The purpose of this work is to consider counterexamples and conditions for the monotonicity of a function in analysis.
To achieve the goal, the following tasks were set:
1. Consider counterexamples in the analysis
2. Define the concept of a counterexample
3. Consider the use of counterexamples in differentiation
4. Define the concept of monotonicity of functions
5. Characterize the conditions for the monotonicity of a function
6. Consider the necessary condition for a local extremum
7. Consider sufficient conditions local extremum
1. Counterexamples in analysis
1.1. The concept of a counterexample
The popular expressions: “learn from examples”, “the power of example” have not only everyday meaning. The word “example” has the same root as the words “measure”, “to measure”, “to measure”, but that is not the only reason why it has been present in mathematics from its very beginnings. An example illustrates the concept, helps to understand its meaning, confirms the truth of the statement in its particular manifestation; a counterexample, refuting a false statement, has evidentiary force.
A counterexample is an example that refutes the truth of a certain statement.
Constructing a counterexample is a common way to refute hypotheses. If there is a statement like “For any X from the set M, property A holds,” then a counterexample to this statement would be any object X 0 from the set M for which property A does not hold.
Classic counterexample in history mathematical analysis is a function constructed by Bernard Bolzano that is continuous on the entire real axis and not differentiable at any point. This function served as a counterexample to the hypothesis that differentiability of a function is a natural consequence of its continuity.
2.2. Using Counterexamples in Differentiation
This section was chosen due to the fact that differentiation is a basic element of mathematical analysis.
In some of the examples in this chapter, the term derivative will also apply to infinite limits.
However, the term differentiable function is used only if the function has a finite derivative at every point in its domain of definition. A function is said to be infinitely differentiable if it has a (finite) derivative of any order at every point in its domain.
An exponential function with base e will be denoted by the symbol ex x or exp(x).
It is assumed that all sets, including domains and value sets of functions, are subsets of R. Otherwise, appropriate clarification will be made.
1. Non-derivative function
Function sgnA: and in general, any function with a discontinuity in the form of a jump does not have a primitive, i.e., is not a derivative of any function, since it does not have the Cauchy property of accepting all intermediate values, and this property is inherent not only in continuous functions, but also in derivatives ( see, p. 84, ex. 40, and also, vol. I, p. 224). Below is an example of a discontinuous derivative.
2. Differentiable function with discontinuous derivative
Consider the function
Its derivative
discontinuous at the point x = 0.
3. A discontinuous function that has a derivative everywhere (not necessarily finite)
In order for such an example to become possible, the definition of derivative must be expanded to include ± values. Then discontinuous function sgn x (example 1) has a derivative
4. Differentiable function whose derivative does not preserve sign in any one-sided neighborhood of the extremal point
has an absolute minimum at the point x = 0. And its derivative
in any one-sided neighborhood of zero takes both positive and negative values. The function f is not monotonic in any one-sided neighborhood of the point x = 0.
5. A differentiable function whose derivative is positive at some point, but the function itself is not monotonic in any neighborhood of this point
has a derivative equal to
In any neighborhood of zero, the derivative f/(x) has both positive and negative values.
6. A function whose derivative is finite, but not limited on a closed interval
Consider the function
Its derivative
not limited to [-1, 1].
7. A function whose derivative exists and is limited, but does not have an (absolute) extremum on a closed interval
has a derivative
In any neighborhood of zero, this derivative has values arbitrarily close to 24 and -24. On the other hand, for 0 Therefore, from the inequality 0< h
1 следует, что 8. Everywhere continuous, but nowhere differentiable function Function | x | is continuous everywhere, but not differentiable at the point x - 0. Using a shift of this function, one can define an everywhere continuous function that is not differentiable at every point of an arbitrarily given finite set. In this section we will give an example using an infinite number of shifts of the function | x |. Let us show that the function is not differentiable anywhere. Let a be an arbitrary real number, and let for every natural number n the number h n equal to 4 -n or –4 -n be chosen so that Then the quantity has the same value | h n | for all m n and equal to zero for m > n. Then the difference ratio is an integer that is even when n is even and odd when n is odd. It follows that the limit does not exist, and therefore does not exist and The example given is a modification of an example constructed by B. L. Van der Waerden in 1930 (see, p. 394). The first example of a continuous, nowhere differentiable function was constructed by K. W. T. Weierstrass (German mathematician, 1815-1897): where a is an integer odd number, and the number b is such that Currently, examples of continuous functions are known that do not even have a one-sided finite or infinite derivative at any point. These examples and further references can be found in (pp. 392-394), (pp. 61 - 62, 115, 126), and (Vol. II, pp. 401-412). The function of the present example is not monotonic on any interval. Moreover, there is an example of a function that is differentiable everywhere and is not monotone anywhere (see, vol. II, pp. 412-421). The construction of this example is very complex and leads to a function that is everywhere differentiable and has a dense set of relative maxima and a dense set of relative minima. 9. Differentiable function for which the mean value theorem does not hold In this example, we are again forced to resort to a complex-valued function. Function of a real variable x is everywhere continuous and differentiable (see, pp. 509-513). However, there is no such interval for which, at some value, the equality If we assume that this equality is possible, then by equating the squares of the modules (absolute values) of both its parts, we obtain the equality which after elementary transformations takes the form But since there is no positive number h such that sin h = h (see, page 78), we get a contradiction. 13. An infinitely differentiable monotone function f such that If monotonicity is not required, then a trivial example of such a function would be, for example, (sinx 2)/x. Let us construct an example of a monotone function that has the indicated property. Let f(x) be equal to 1 for and equal on closed intervals for On the remaining intermediate intervals of the form, we determine f(x) using the function applying horizontal and vertical shifts and multiplication by appropriate negative factors. 2.1. Monotony of functions A function f (x) is said to be increasing on the interval D if for any numbers x 1 and x 2 from the interval D such that x 1< x 2 , выполняется неравенство
f (x 1) < f (x 2). A function f (x) is said to be decreasing on the interval D if for any numbers x 1 and x 2 from the interval D such that x 1< x
2 , выполняется неравенство f (x 1) >f(x2). Figure 1. In the graph shown in the figure, the function y = f (x) increases on each of the intervals [ a ; x 1) and (x 2 ; b ] and decreases on the interval (x 1 ; x 2). Note that the function increases on each of the intervals [ a ; x 1) and (x 2 ; b ], but not on the union gaps If a function increases or decreases on a certain interval, then it is called monotonic on this interval. Note that if f is a monotone function on the interval D (f (x)), then the equation f (x) = const cannot have more than one root on this interval. Indeed, if x 1< x 2 – корни этого
уравнения на промежутке D (f (x)), то f (x 1) = f (x 2) = 0, что противоречит
условию монотонности. Let's list the properties monotonic functions(it is assumed that all functions are defined on some interval D). Similar statements can be formulated for a decreasing function. Rice. 2. Properties of the function. A point a is called a maximum point of a function f if there is an ε-neighborhood of the point a such that for any x in this neighborhood the inequality f (a) ≥ f (x) holds. A point a is called a minimum point of a function f if there is an ε-neighborhood of the point a such that for any x in this neighborhood the inequality f (a) ≤ f (x) holds. The points at which the maximum or minimum of the function is reached are called extremum points. At the extremum point, the nature of the monotonicity of the function changes. Thus, to the left of the extremum point the function can increase, and to the right it can decrease. According to the definition, the extremum point must be internal point domain of definition. If for any (x ≠ a) the inequality f (x) ≤ f (a) holds, then point a is called the point of the largest value of the function on the set D: If for any (x ≠ b) the inequality f (x) > f (b) is satisfied, then point b is called the point of the minimum value of the function on the set D. Complex function Weierstrass looks like where - some real number, but is written either as , or as . The real and imaginary parts of the function are called the cosine and Weierstrass sinusoids, respectively. The function is continuous, but nowhere differentiable. However, its formal generalization to the case is both continuous and differentiable. In addition to the function itself, this section discusses some of its options; the need for their presentation is due to the new meaning that the theory of fractals gave to the Weierstrass function. Frequency spectrum of a function. The term “spectrum”, in my opinion, is overloaded with meanings. The frequency spectrum refers to the set acceptable values frequencies without regard to the amplitudes of the corresponding components. The frequency spectrum of a periodic function is a sequence of positive integers. The frequency spectrum of the Brownian function is . The frequency spectrum of the Weierstrass function is a discrete sequence from to . Energy spectrum of a function. The sub-energy spectrum is understood as the set of permissible frequency values together with the energy values (squared amplitudes) of the corresponding components. For each frequency value of the form in the function there is a spectral line of energy of the form Comparison with fractional Brownian motion. The total energy is proportional in several other cases we considered earlier: fractional periodic random Fourier–Brown–Wiener functions, the permissible frequencies for which have the form , and the corresponding Fourier coefficients are equal to ; random processes with a continuous spectral population density proportional to . Latest processes are nothing more than fractional Brownian functions, described in Chapter 27. For example, at one can detect the cumulative spectrum of the Weierstrass function in ordinary Brownian motion, the spectral density of which is proportional to . A significant difference: the Brownian spectrum is absolutely continuous, while the spectra of the Fourier–Brown–Wiener and Weierstrass functions are discrete. Non-differentiability. To prove that a function does not have a finite derivative for any value, Weierstrass had to combine two following conditions: is an odd integer, as a result of which the function is a Fourier series, and . We took the necessary and sufficient conditions ( and ) from Hardy's article. Energy consumption. To a physicist accustomed to spectra, Hardy's conditions seem obvious. Applying the empirical rule that the derivative of a function is calculated by multiplying its Fourier coefficient by , the physicist finds for the formal derivative of the function that the squared amplitude of the Fourier coefficient c is equal to It is interesting to note that Riemann, in search of an example of non-differentiability, came up with the function Ultraviolet divergence/catastrophe. The term "catastrophe" appeared in physics in the first decade of the twentieth century, when Rayleigh and Jeans independently developed the theory of blackbody radiation, according to which the energy of the frequency range of width in the vicinity of the frequency is proportional to . This means that the total energy of the spectrum is high frequencies infinite - which turns out to be very catastrophic for the theory. Since the source of trouble comes from frequencies beyond the ultraviolet part of the spectrum, the phenomenon is called an ultraviolet (UV) catastrophe. Everyone knows that Planck built his quantum theory on the ruins into which the UV catastrophe turned the theory of radiation. Historical retreat. Let us note (although I do not quite understand why no one has done this before; in any case, I have not found anything similar in the sources available to me) that the cause of death of both old physics and old mathematics is the same divergence that undermined them the belief that continuous functions simply must be differentiable. Physicists reacted simple change rules of the game, mathematicians had to learn to live with non-differentiable functions and their formal derivatives. (The latter is the only example of a generalized Schwarz function often used in physics.) In search of a scale-invariant discrete spectrum. Infrared divergence. Although frequency spectrum Brownian function is continuous, scale-invariant and exists at , the frequency spectrum of the Weierstrass function corresponding to the same value is discrete and limited from below by the value . The presence of the lower limit is due solely to the fact that Weierstrass’s number was initially integer, and the function was periodic. To eliminate this circumstance, it should obviously be allowed to take any value from to . And in order for the energy spectrum to become scale-invariant, it is enough to associate each frequency component with an amplitude. Unfortunately, the resulting series diverges, and low-frequency components are to blame. This defect is called infrared (IR) divergence (or “catastrophe”). Be that as it may, we have to put up with this divergence, since otherwise the lower limit comes into conflict with the self-similarity inherent in the energy spectrum. Modified Weierstrass function, self-affine with respect to focal time. The simplest procedure, which allows one to continue the frequency spectrum of the Weierstrass function to a value and avoid catastrophic consequences, consists of two stages: first we obtain the expression is still continuous, but nowhere differentiable. In addition, it is scale-invariant in the sense that So the function Gaussian random functions with the generalized Weierstrass spectrum. The next step towards realism and broad applicability is randomization of the generalized Weierstrass function. The simplest and most natural method consists in multiplying its Fourier coefficients by independent complex Gaussian random variables with zero mathematical expectation and unit variance. The real and imaginary parts of the resulting function can rightfully be called Weierstrass–Gauss (modified) functions. In some senses, these functions can be considered approximate fractional Brownian functions. When the values coincide, their spectra are as similar as the fact that one of these spectra is continuous and the other discrete allows. Moreover, the results of Orey and Marcus (see p. 490) are applicable to the Weierstrass–Gauss functions, and the fractal dimensions of their level sets coincide with the fractal dimensions of the level sets of fractional Brownian functions. Considering the precedent represented by the fractional Brownian motion, we can assume that the dimension of the zero-sets of the Weierstrass–Rademacher function will be equal to . This assumption is confirmed in , but only for integers. Singh mentions many other variations of the Weierstrass function. The dimension zero of the sets of some of them is easy to estimate. In general, this topic clearly deserves more detailed study, taking into account the achievements of modern theoretical thought. Let's construct an auxiliary function on a segment step by step. At the zero step we will set two points: Next we fix the parameter. At the first and subsequent steps we will set points according to next rule: for every two previously constructed points adjacent to the x-axis and we will construct two new points and centrally symmetrically with respect to the center of the rectangle defined by the points and with a coefficient k. That is, at the first step two new points are specified: On (m+1)- om step in addition to the previously constructed points with abscissas two points are constructed in all spaces along the x-axis between adjacent already constructed points. This construction is carried out as follows: the spaces along the abscissa axis between adjacent points (rectangles with sides a And b) are divided into 3 equal parts each. Then two new points are constructed according to one of the following schemes: Depending on which of the neighboring points is higher or higher, we use the left or right scheme. At the first step, as shown above, we accept a = b = 1. We repeat the construction a countable number of times for m = 1, 2, 3, …. As a result, we will obtain a fractal that will be similar, up to a certain affine transformation(stretching, compression, rotation) of any of its parts contained in each strip: As a result of constructing a fractal, we obtain a function defined on a set of points which is dense everywhere on the segment . What properties does the constructed function have? · at each point of the form (*) there is either a strict maximum or a strict minimum, i.e. function g(x) is nowhere monotonic, and has dense sets of strict extrema points on the segment; · the function g(x) is continuous, and even uniformly continuous on the set of points (*); · the function constructed continuous on the segment does not have at any point of this segment even one-sided derivatives; The above properties were proven in the course “Selected Chapters of Mathematical Analysis”. In the example considered, we assumed the parameter . By changing the value of this parameter, you can obtain families of functions with their own special properties. · . These functions are continuous and strictly monotonically increasing. They have zero and infinite derivatives (respectively, inflection points) on sets of points that are dense everywhere on the segment. · . Received linear function y = x · . The properties of the family of functions are the same as for values of k from the first range. · . We have obtained the Cantor function, which was studied in detail by us earlier. · . These functions are continuous, nowhere monotone, have strict minima and maxima, zero and infinite (of both signs) one-sided derivatives on sets of points that are dense everywhere on the segment. · . This function was studied by us above. · . Functions from this range have the same properties as the function at . Conclusion. In my work, I implemented some examples from the course “Selected Chapters of Mathematical Analysis”. IN this work Screenshots of the programs I visualized were inserted. In fact, they are all interactive; the student can see the view of the function on specific step, build them yourself iteratively and bring the scale closer. Construction algorithms, as well as some library functions Skeleton were specially selected and improved for this type problems (mainly fractals were considered). This material will undoubtedly be useful to teachers and students and is a good accompaniment to the lectures of the course “Selected Chapters of Mathematical Analysis”. The interactivity of these visualizations helps to better understand the nature of the constructed sets and facilitate the process of perception of the material by students. The described programs are included in the library of visual modules of the project www.visualmath.ru, for example, here is the Cantor function we have already considered: In the future, it is planned to expand the list of visualized tasks and improve construction algorithms for more efficient work programs. Working in the www.visualmath.ru project undoubtedly brought a lot of benefit and experience, teamwork skills, the ability to evaluate and present educational material as clearly as possible. Literature. 1. B. Gelbaum, J. Olmsted, Counterexamples in analysis. M.: Mir.1967. 2. B.M. Makarov et al. Selected problems in real analysis. Nevsky dialect, 2004. 3. B. Mandelbrot. Fractal geometry of nature. Institute for Computer Studies, 2002. 4. Yu.S. Ochan, Collection of problems and theorems on TFDP. M.: Enlightenment. 1963. 5. V.M. Shibinsky Examples and counterexamples in the course of mathematical analysis. M.: graduate School, 2007. 6. R.M. Kronover, Fractals and chaos in dynamic systems, M.: Postmarket, 2000. 7. A. A. Nikitin, Selected chapters of mathematical analysis // Collection of articles by young scientists of the Faculty of Computational Mathematics and Mathematics of Moscow State University, 2011 / ed. S. A. Lozhkin. M.: Publishing department of the Faculty of Computational Mathematics and Mathematics of Moscow State University. M.V. Lomonosova, 2011. pp. 71-73. 8. R.M. Kronover, Fractals and chaos in dynamic systems, M.: Postmarket, 2000. 9. Fractal and construction of an everywhere continuous, but nowhere non-differentiable function // XVI International Lomonosov Readings: Collection scientific works. – Arkhangelsk: Pomeranian State University, 2004. P.266-273. The union of a countable number of open sets (adjacent intervals) is open, and the complement to an open set is closed. Any neighborhood of a point A Cantor set, there is at least one point from , different from A. Closed and does not contain isolated points(each point is a limit). There is at most a countable set that is everywhere dense in . A set A is nowhere dense in the space R if any open set of this space contains another open set, completely free from points of set A. A point, any neighborhood of which contains an uncountable set of points of a given set. We will say that a set on a plane is nowhere dense in metric space R, if any open circle of this space contains another open circle, completely free from points of this set. Let's construct an interesting set on the plane IN as follows: divide, square by straight lines The Sierpinski Cemetery is a perfect and nowhere dense multitude. Let us note the fractal structure of the set. Let's call Cantor comb many D on the plane Oxy, consisting of all points Is it possible to set B(Sierpinski Cemetery) and D(Cantor comb) express through the Cantor set B= D= x Is it possible to map continuously a set that is nowhere dense on a segment onto this segment itself? Yes, let’s take the Cantor set, which is not dense anywhere. At the first step of construction, we set the value of the function equal to 0.5 at the points of the adjacent interval of the first kind. At the second step, for each adjacent interval of the second kind we assign the function value 0.25 and 0.75, respectively. Those. we seem to divide each segment into an axis Oy in half ( y i) and set in the corresponding adjacent interval the value of the function equal to the value yi. As a result, we received a non-decreasing function (it was proven in the course “Selected Chapters of Mathematical Analysis”), defined on the segment and constant in a certain neighborhood of each point from the set \ Note the fractal structure of the function: Function The Cantor function is continuous on the interval. It does not decrease by and the set of its values constitutes the entire segment. Therefore, the function An interesting observation is that the graph of the continuous Cantor function Let's build an auxiliary function Next we fix the parameter On (m+1)- om step in addition to the previously constructed points with abscissas two points are constructed in all spaces along the abscissa between adjacent already constructed points. This construction is carried out as follows: the gaps along the abscissa axis between adjacent points (rectangles with sides a And b) are divided into 3 equal parts each. Then two new points are constructed according to one of the following schemes: Depending on which of the neighboring points We repeat the construction a countable number of times for m = 1, 2, 3, …. As a result, we will obtain a fractal that will be similar, up to some affine transformation (stretching, compression, rotation) of any of its parts contained in each strip: As a result of constructing the fractal, we obtain the function which is dense everywhere on the segment . What properties does the constructed function have? at each point of the form (*) there is either a strict maximum or a strict minimum, i.e. function g(x)
is not monotonic anywhere, and has dense sets of strict extrema points on the segment; the function g(x) is continuous, and even uniformly continuous on the set of points (*); the function constructed continuous on a segment does not have even one-sided derivatives at any point of this segment; The above properties were proven in the course “Selected Chapters of Mathematical Analysis”. In the example considered, we assumed the parameter 2. Monotonic functions
. Consequently, the total energy value at frequencies converges and is proportional to 
.
. Since the total energy at frequencies greater than θ is infinite, it becomes clear to physicists that the derivative cannot be determined.
, the spectral energy of which at frequencies higher than , is proportional to , where . Thus, using the same heuristic reasoning, one can assume that the derivative is non-differentiable. This conclusion is only partially true, since at certain values the derivative still exists (see).
, and only then allow it to take any value from to . The additional terms corresponding to the values converge, and their sum is continuous and differentiable. The function modified in this way
.
does not depend on . We can say it differently: with function 
does not depend on . That is, the function 
, its real and imaginary parts are self-affine with respect to the values of the form and focal time.
And 
.
And 
, etc.
,
; 
by 9 equal squares and throw out five of them open, not adjacent to the vertices of the original square. Then, we also divide each of the remaining squares into 9 parts, and discard five of them, etc. The set remaining after a countable number of steps is denoted by B and let's call Sierpinski Cemetery. Let's calculate the area of the discarded squares:
2.2 Cantor's comb
, whose coordinates satisfy the following conditions: 
, Where 
- Cantor set on the axis Oy. A Cantor comb is a perfect nowhere dense set on the plane. Many D consists of all points 
original unit square, whose abscissas are arbitrary 
, and the ordinates can be written as a ternary fraction that does not contain a unit among its ternary signs.
using the operations of the segment's complement and the Cartesian product? It is obvious that the sets B And D expressed simply:
x 
3 Cantor function
. Constructed function 
called Cantor function(Cantor function), and its graph below is ""devil's stairs"".
satisfies the following inequality:
has no jumps. And because a monotone function cannot have other discontinuity points other than jumps (see the criterion for the continuity of monotone functions), then it is continuous.
It is impossible to draw "without lifting the pencil from the paper."A function that is continuous everywhere but differentiable nowhere
on a segment step by step. At the zero step we will set two points:
And 
.
. At the first and subsequent steps, we will specify points according to the following rule: for every two previously constructed points adjacent to the abscissa axis 
And 
we will build two new points 
And 
centrally symmetrical relative to the center of the rectangle defined by the points 
And 
with coefficient k. That is, at the first step two new points are specified:
And 
, etc.
,
or 
above, use the left or right scheme. At the first step, as shown above, we accept a = b = 1.
;
, defined on a set of points
,
;
(*)
. By changing the value of this parameter, you can obtain families of functions with their own special properties.
