MINISTRY OF EDUCATION AND SCIENCE OF THE RUSSIAN FEDERATION
STATE GENERAL EDUCATIONAL INSTITUTION
HIGHER PROFESSIONAL EDUCATION
"USSURIYSK STATE PEDAGOGICAL INSTITUTE"
Faculty of Physics and Mathematics
Coursework in calculus
Topic: "Continuous but not differentiable functions"
Completed by: Ksenia Plyasheshnik
student of group 131
Head: Delyukova Ya.V.
Ussuriysk - 2011
Introduction ................................................ ............................................. 3
History reference................................................ ......................... four
Basic definitions and theorems ............................................................... ....... 5
An example of a continuous function without a derivative .............................. 10
Exercise solution .................................................................. ........................ 13
Conclusion................................................. ............................................... 21
Bibliography................................................ ........................... 22
Introduction
Course work is devoted to the study of the relationship between continuity and the existence of a derivative of a function of one variable. Based on the goal, the following tasks were set:
1. Study educational literature;
2. Study an example of a continuous function that does not have a derivative at any point, built by van der Waerden;
3. Solve the system of exercises.
History reference
Bartel Leendert van der Waerden (Dutch. Bartel Leendert van der Waerden, February 2, 1903, Amsterdam, the Netherlands - January 12, 1996, Zurich, Switzerland) - Dutch mathematician.
He studied at the University of Amsterdam, then at the University of Göttingen, where he was greatly influenced by Emmy Noether.
His main works are in the field of algebra, algebraic geometry, where he (along with André Weil and O. Zarissky) raised the level of rigor, and mathematical physics, where he was engaged in the application of group theory to questions of quantum mechanics (along with Herman Weil and J. Wigner). His classic book Modern Algebra (1930) became the model for later textbooks on abstract algebra and went through many reprints.
Van der Waerden is one of the greatest specialists in the history of mathematics and astronomy in the ancient world. His Awakening Science (Ontwakende wetenschap 1950, Russian translation 1959) gives a detailed account of the history of mathematics and astronomy in ancient Egypt, Babylon and Greece. The Appendix to the Russian translation of this book contains the article "The Pythagorean Doctrine of Harmony" (1943) - a fundamental exposition of the Pythagorean views on musical harmony.
Basic definitions and theorems
Limit of a function at a point. Left and right limits
Definition (limit according to Cauchy, in the language Number is called the limit of a function at a point, if
Definition (in the language of the neighborhood) A number is called the limit of a function at a point if for any -neighborhood of the number there exists -neighborhood of the point such that as soon as 
Definition (according to Heine) A number is called the limit of a function at a point if for any sequence converging to (i.e., the corresponding sequence of values of the function converges to the number 
Definition A number is called the left limit of a function at a point if
Definition A number is called the right limit of a function at a point if
Theorem (necessary and sufficient condition for the existence of a limit)
In order for a function limit to exist at a point, it is necessary and sufficient that the left and right limits exist equal to each other.
The concept of a derivative. One-sided derivatives.
Consider a function defined on the set
1. Let's take the increment . Let's increment the point. Get .
2. Let's calculate the value of the function in points . and
3. .
4. .
moreover, the increment of the argument can be both positive and negative, then this limit is called the derivative at the point and is denoted by . It can also be endless.
the left (left-side) derivative of the function at the point , and if
there is a finite limit 
then it is called the right-hand derivative of the function at the point .
A function has at a point if and only if its left and right derivatives coincide at a point:
( ( .
Consider the function 
Find one-sided derivatives at a point
Consequently, ( =-1; ( =1 and ( ( , that is, at a point, the function has no derivative.
Various definitions of the continuity of a function at a point.
Definition 1 (basic) A function is called continuous at a point if the limit of the function at is equal to the value of the function at this point.
Definition 2 (in the language A function is called continuous at a point if ε, δ>0, such that .
Definition 3 (according to Heine, in the language of sequences) A function is called continuous at a point if for any sequence converging to a point the corresponding sequence of function values converges to .
Definition 4 (in the language of increments) A function is called continuous at a point if an infinitesimal increment of the argument corresponds to an infinitesimal increment of the function.
The concept of a differentiable function
Definition 1 A function defined on a set (is called differentiable at a point if its increment at this point can be represented as (*), where A - const , independent of , - infinitesimal at
Definition 2 A function that is differentiable at any point of a set is called differentiable on the set.
Relationship between differentiability and continuity
Theorem. If a function is differentiable at a point , then it is continuous at a point .
Proof.
Let a function be given The function is differentiable at the point , where
Inverse theorem. If a function is continuous, then it is differentiable.
The converse theorem is not true.
B is not differentiable, although it is continuous.
Classification of break points
Definition A function that is not continuous at a point is discontinuous at a point , and the point itself is called a discontinuity point.
There are two classifications of discontinuity points: I and II kind.
Definition A point is called a discontinuity point of the first kind if at this point there are finite one-sided limits that are unequal to each other.
Definition A point is called a disposable point yva if , but they are not equal to the value of the function at point .
Definition A point is called a discontinuity point of the second kind if at this point the one-sided limits are equal, or one of the one-sided limits is infinite, or there is no limit at the point.
· – endless;
· – endless or – endless;
Signs of uniform convergence of a series in
Weierstrass sign.
If the members of the functional series (1) satisfy the inequalities in the region 
where is a member of some convergent numerical series, then the series (1) converges to uniformly.
Theorem 1 Let the functions 
are defined in an interval and are all continuous at some point in this interval. If the series (1) in the interval converges uniformly, then the sum of the series at the point will also be continuous.
An example of a continuous function without a derivative
The first example of this kind was built by Weierstrass; its function is defined next:
where 0< a <1, а b есть нечетное натуральное число (причем ab >1+π). This series is majorized by a convergent progression, therefore (signs of uniform convergence of series), converges uniformly, and its sum is an everywhere continuous function of x. Through painstaking research, Weierstrass managed to show that, nevertheless, at no point does it have a finite derivative.
Here we will consider a simpler example of van der Waerden, built essentially on the same idea, only the oscillating curves y = cosωχ are replaced by oscillating broken lines.
So, we denote by the absolute value of the difference between the number χ and the nearest integer to it. This function will be linear in each interval of the form , where s is an integer; it is continuous and has a period of 1. Its graph is a broken line, it is shown in Fig. 1; individual links of the polyline have a slope of ±1.
Let us then, for k=1,2,3,…:
This function will be linear in intervals of the form ; it is also continuous and has period . Her graph is also broken, but with smaller teeth; Fig. 1(b), for example, shows a graph of the function . In all cases, the slope coefficients of individual links of the polyline and here are equal to ±1.
Let us now define, for all real values x , the function f (x) by the equality
Since, obviously, 0≤ (k =0,1,2,…), so that the series is dominated by a convergent progression , then (as in the case of the Weierstrass function) the series converges uniformly, and the function is everywhere continuous.
Let's stop at any value. Calculating it with an accuracy up to (where n = 0,1,2, ...), by deficiency and by excess, we will conclude it between numbers of the form:
≤ , where is an integer.
(n=0,1,2,…).
It is obvious that closed intervals turn out to be nested one into another. In each of them there is such a point that its distance from the point is equal to half the length of the interval.
It is clear that as n increases, the variant .
Let us now compose the ratio of increments
=
But when k > n , the number is an integer multiple of the periods of the function , the corresponding terms of the series turn to 0 and can be omitted. If k ≤ n , then the function , which is linear in the interval , will be linear in the interval contained on it , and
(k=0,1,…,n).
Thus, we finally have 
in other words, this ratio is equal to an even integer when n is odd and an odd number when n is even. From this it is clear that at , the ratio of increments cannot tend to any finite limit, so our function does not have at a finite derivative.
Exercise Solution
Exercise 1 (, #909)
The function is defined as follows: . Investigate continuity and find out existence
Na is continuous as a polynomial;
On (0;1) 
is continuous as a polynomial;
On (1;2) is continuous as a polynomial;
On (2; is continuous as an elementary function.
Points suspicious for breaking
Since the left limit is equal to the right limit and is equal to the value of the function at the point, the function is continuous at the point
Since the left limit is equal to the value of the function at the point, the function is discontinuous at the point .
1 way. There is no finite derivative of the function at a point. Indeed, suppose the contrary. Let there be a finite derivative of the function at a point 
is continuous at a point (by Theorem 1: If a function is differentiable at a point, then it is continuous.
2 way. Let's find the one-sided limits of the function at the point x =0.
Exercise 2 (, №991)
Show that function 
has a discontinuous derivative.
Let's find the derivative of the function.
The limit does not exist discontinuous at a point
Since is an infinitesimal function, it is bounded.
Let us prove that the function 
has no limit at the point.
To prove it, it suffices to show that there are two sequences of argument values converging to 0, which does not converge to
Output: function 
has no limit at the point.
Exercise 3 (, #995)
Show that the function where is a continuous function and has no derivative at the point . What are one-sided derivatives
One-sided limits are not equal the function does not have a derivative at the point .
Exercise 4 (, #996)
Construct an example of a continuous function that does not have a function derivative at given points:
Consider a function at points
Let's find one-sided limits
One-sided limits are not equal the function does not have a derivative at the point . Similarly, the function has no derivatives at other points
Exercise 5 (, №125)
Show that the function has no derivative at .
Let's find the increment of the function at the point
Compose the ratio of the function increment at a point to the argument increment
Let's go to the limit
Exercise 6 (, №128)
Show that function 
has no derivative at the point .
Let's take an increment Let's give an increment to the point Get
Find the value of the function at points and
Let's find the increment of the function at the point
Compose the ratio of the function increment at a point to the argument increment
Let's go to the limit
Conclusion: does not have a finite derivative at the point .
Exercise 7 (, №131)
Investigate a Function for Continuity
- a point suspicious for a break
Since the left limit is equal to the value of the function at the point, the function is continuous at the point, there is a discontinuity of the first kind.
Conclusion
The term paper presents the material related to the concept of "Continuous but not differentiable functions", the goals of this work have been achieved, the tasks have been solved.
Bibliography
1. B. P. Demidovich, / Collection of problems in the course of mathematical analysis. Textbook for students of the Faculty of Physics and Mathematics of Pedagogical Institutes. - M.: Enlightenment, 1990 -624s.
2. G. N. Berman, / Collection of problems in the course of mathematical analysis. - M.: Nauka, 1977 - 416s.
3. G. M. Fikhtengolts, / Course of differential and integral calculus vol. II. - M., Nauka, 1970 - 800s.
4. I.A. Vinogradova, / Tasks and exercises in mathematical analysis, part 1. - M.: Bustard, 2001 - 725s.
5. Internet resource \ http://ru.wikipedia.org/wiki.
6. Internet resource \http://www.mathelp.spb.ru/ma.htm.
Let us construct an interesting set on the plane AT as follows: divide, square by straight lines 
into 9 equal squares and discard 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, and so on. The set remaining after a countable number of steps is denoted B and call Sierpinski cemetery. Calculate the area of the discarded squares:
The Sierpinski cemetery is perfect and nowhere dense.
Note the fractal structure of the set.
2.2 Cantor's comb
Let's call Cantor comb lots of D on surface Oxy, consisting of all points 
, whose coordinates satisfy the following conditions: 
, where 
- Cantor set on the axis Oy. The Cantor comb is a perfect nowhere dense set in the plane. Lots of D consists of all points 
the original unit square, the abscissas of which are arbitrary 
, and the ordinates can be written as a ternary fraction that does not contain one among its ternary signs.
Is it possible to set B(cemetery of Sierpinski) and D(Cantor comb) express in terms of the Cantor set 
with the help of operations of the addition to the segment and the Cartesian product? It is obvious that the sets B and D are expressed simply:
B=
x 
D= x 
3 Cantor function
Is it possible to continuously map some nowhere dense set on a segment onto this segment itself?
Yes, let's take Cantor's nowhere dense set. At the first step of the 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 set the value of the function to 0.25 and 0.75, respectively. Those. we, as it were, 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 have obtained a non-decreasing function (which was proved within the framework of the course "Selected Chapters of Mathematical Analysis"), defined on a segment and constant in some neighborhood of each point from the set \ 
. Built function 
called Cantor function(Cantor function), and its graph below - "Damn Stairs".
Pay attention to the fractal structure of the function:
Function 
satisfies the following inequality:
The Cantor function is continuous on the interval . It does not decrease on and the set of its values makes up the entire segment . Therefore, the function 
has no jumps. And since If a monotone function cannot have other points of discontinuity than jumps (see the criterion for the continuity of monotone functions), then it is continuous.
Curious is the observation that the graph of the continuous Cantor function 
it is impossible to draw "without lifting the pencil from the paper"".
Everywhere continuous but nowhere differentiable function
Let's build an auxiliary function 
on the step by step. At the zero step, we set two points:
and 
.
Next, fix the parameter 
. At the first and subsequent steps, we will specify points according to the following rule: for each two previously constructed points adjacent along the abscissa axis 
and 
we will build two new points 
and 
centrally symmetrical about the center of the rectangle defined by the points 
and 
with coefficient k. That is, at the first step, two new points are set:
and 
, etc.
On the (m+1)- th step in addition to the previously constructed points with abscissas
,
two points are built in all intervals along the abscissa axis between neighboring already built points. This construction is performed as follows: the gaps along the abscissa between adjacent points (rectangles with sides a and b) are divided into 3 equal parts each. Then two new points are built according to one of the following schemes:
Depending on which of the neighboring points 
or 
above, use the left or right scheme. At the first step, as shown above, we take a=b=1.
We repeat the construction a countable number of times for m = 1, 2, 3, … . As a result, we will get a fractal that will be similar, up to some affine transformation (expansion, compression, rotation) of any of its parts contained in each strip:
;
As a result of constructing the fractal, we obtain the function 
defined on the set of points
,
;
(*)
which is everywhere dense on the segment .
What properties does the constructed function have?
at each point of the form (*) either a strict maximum or a strict minimum, i.e. function g(x) nowhere monotonic, and has dense sets of points of strict extrema 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 even one-sided derivatives at any point of the given segment;
The above properties were proved within the framework of the course "Selected Chapters of Mathematical Analysis".
In the considered example, we set the parameter 
. By changing the value of this parameter, you can get families of functions with their own special properties.
The complex Weierstrass function has the form
where is some real number, and is written either as , or as . The real and imaginary parts of a function are called the Weierstrass cosine and 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 representation is due to the new meaning given to the Weierstrass function by the theory of fractals.
The frequency spectrum of the function. The term "spectrum", in my opinion, is overloaded with meanings. The frequency spectrum is understood as the set of permissible frequency values, regardless of 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 the function. The energy spectrum is understood as the set of allowable frequency values together with the energy values (amplitude squares) of the corresponding components. For each frequency value of the form in the function, there is a spectral line of energy of the form 
. Therefore, the total value of the energy at frequencies converges and is proportional to 
.
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 admissible frequencies for which have the form , and the corresponding Fourier coefficients are equal to ; random processes with a continuous spectral population density proportional to . The latter processes are nothing but the fractional Brownian functions described in Chapter 27. For example, at , one can find the cumulative spectrum of the Weierstrass function in an ordinary Brownian motion whose spectral density is proportional to . The essential difference is that the Brownian spectrum is absolutely continuous, while the spectra of the Fourier–Brown–Wiener and Weierstrass functions are discrete.
Non-differentiability. To prove that the function does not have a finite derivative for any value, Weierstrass had to combine the following two conditions: - an odd integer, as a result of which the function is a Fourier series, and . Necessary and sufficient conditions ( and ) are taken from Hardy's article.
Energy consumption. To a physicist accustomed to spectra, Hardy's conditions seem obvious. Applying the rule of thumb that the derivative of a function is calculated by multiplying its -th Fourier coefficient by , the physicist finds for the formal derivative of the function that the square of the amplitude of the Fourier coefficient c is 
. Since the total energy at frequencies greater than , is infinite, it becomes clear to the physicist that the derivative cannot be determined.
It is interesting to note that Riemann, in search of an example of non-differentiability, came to the function 
, whose spectrum energy at frequencies greater than , is proportional to , where . Thus, applying the same heuristic reasoning, we can assume that the derivative is non-differentiable. This conclusion is only partly true, since for certain values the derivative still exists (see ).
Ultraviolet divergence / catastrophe. The term "catastrophe" appeared in physics in the first decade of the 20th century, when Rayleigh and Jeans independently developed the theory of black body radiation, according to which the energy of the frequency range of the width in the vicinity of the frequency is proportional to . This means that the total energy of the spectrum at high frequencies is infinite - which turns out to be quite catastrophic for the theory. Since the source of trouble is the frequencies lying beyond the ultraviolet part of the spectrum, the phenomenon is called the ultraviolet (UV) catastrophe.
Everyone knows that Planck built his quantum theory on the ruins into which the theory of radiation was turned precisely by the UV catastrophe.
Historical retreat. We note (although I do not quite understand why no one has done this before; in any case, I did not find 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 have to be differentiable. Physicists reacted by simply changing the rules of the game, while mathematicians had to learn to live with non-differentiable functions and their formal derivatives. (The latter is the only example of a generalized Schwartz function commonly used in physics.)
In search of a scale-invariant discrete spectrum. Infrared divergence. Although the frequency spectrum of the Brownian function is continuous, scale-invariant, and exists for , the frequency spectrum of the Weierstrass function, corresponding to the same value of , is discrete and bounded below by . The presence of the lower bound is due solely to the fact that the Weierstrass number was originally an 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 the amplitude .
Unfortunately, the resulting series diverges, and the low-frequency components are to blame. Such a defect is called infrared (IR) divergence (or "catastrophe"). Be that as it may, one has to put up with this divergence, since otherwise the lower bound conflicts with the self-similarity inherent in the energy spectrum.
Modified Weierstrass function, self-affine with respect to focal time. The simplest procedure for extending the frequency spectrum of the Weierstrass function to a value and avoiding catastrophic consequences in the process consists of two stages: first, we obtain the expression 
, and only then let us take any value from to . The additional terms corresponding to the values for converge, and their sum is continuous and differentiable. Function modified in this way
is still continuous, but nowhere differentiable.
In addition, it is scale-invariant in the sense that
.
So the function 
does not depend on . It can be said differently: for a 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.
Gaussian random functions with generalized Weierstrass spectrum. The next step towards realism and wide applicability is the randomization of the generalized Weierstrass function. The simplest and most natural method is to multiply its Fourier coefficients by independent complex Gaussian random variables with mean zero and unit variance. The real and imaginary parts of the resulting function can rightly be called the Weierstrass-Gauss (modified) functions. In some senses, these functions can be considered approximate fractional Brownian functions. When the values match, 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 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 variants of the Weierstrass function. The zero-dimensionality of the sets of some of them is easily estimable. In general, this topic clearly deserves a more detailed study, taking into account the achievements of modern theoretical thought.
Let's build an auxiliary function on the interval step by step. At the zero step, we set two points:
and 
.
Next, fix the parameter . At the first and subsequent steps, we will specify points according to the following rule: for each two previously constructed points and adjacent along the abscissa axis, we will construct two new points and symmetrically about the center of the rectangle specified by the points and with the coefficient k. That is, at the first step, two new points are set:
and 
, etc.
On the (m+1)- th step in addition to the previously constructed points with abscissas
,
two points are built in all intervals along the abscissa axis between neighboring already built points. This construction is performed as follows: the gaps along the abscissa between adjacent points (rectangles with sides a and b) are divided into 3 equal parts each. Then two new points are built according to one of the following schemes:
Depending on which of the neighboring points or higher, we use the left or right scheme. At the first step, as shown above, we take a=b=1.
We repeat the construction a countable number of times for m = 1, 2, 3, … . As a result, we will get a fractal that will be similar, up to some affine transformation (expansion, 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 everywhere dense 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) nowhere monotonic, and has dense sets of points of strict extrema 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 even one-sided derivatives at any point of the given segment;
The above properties were proved within the framework of the course "Selected Chapters of Mathematical Analysis".
In the considered example, we assumed the parameter . By changing the value of this parameter, you can get 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 everywhere dense on the segment .
· . Linear function obtained y=x
· . The properties of the family of functions are the same as for the values of k from the first range.
· . We have obtained the Cantor function, which we studied in detail 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 everywhere dense on the segment .
· . This function has been studied by us above.
· . Functions from this range have the same properties as the function for .
Conclusion.
In my work, I implemented some examples from the course "Selected Chapters of Mathematical Analysis". Screenshots of programs visualized by me were inserted into this work. In fact, they are all interactive, the student can see the type of function at a particular step, build them iteratively and zoom in. Construction algorithms, as well as some library functions Skeleton were specially selected and improved for this type of problem (mostly 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 www.visualmath.ru project, 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 the construction algorithms for more efficient operation of programs. Working in the www.visualmath.ru project undoubtedly brought a lot of benefits and experience, teamwork skills, the ability to evaluate and present educational material as clearly as possible.
Literature.
1. B. Gelbaum, J. Olmstead, 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 Research, 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. Moscow: Higher school, 2007.
6. R.M. Kronover, Fractals and chaos in dynamical systems, Moscow: Postmarket, 2000.
7. A. A. Nikitin, Selected Chapters of Mathematical Analysis // Collection of articles by young scientists of the faculty of the CMC MSU, 2011 / ed. S. A. Lozhkin. M.: Publishing department of the faculty of the VMK Moscow State University. M.V. Lomonosov, 2011. S. 71-73.
8. R.M. Kronover, Fractals and chaos in dynamical systems, Moscow: Postmarket, 2000.
9. Fractal and the construction of an everywhere continuous but nowhere non-differentiable function // XVI International Lomonosov Readings: Collection of scientific papers. - Arkhangelsk: Pomor 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 in , different from a.
It is closed and does not contain isolated points (each point is a limit point).
There is at most a countable set everywhere dense in .
A set A is nowhere dense in a space R if any open set of this space contains another open set that is completely free of points in A.
A point in any neighborhood of which contains an uncountable set of points of the given set.
We say that a set in the plane is nowhere dense in a metric space R if any open disk in this space contains another open disk that is completely free of points in the given set.
“Is the statement S true?” is perhaps the most typical question in mathematics, when the statement has the form: “Each element of class A also belongs to class B: A B.” To prove that such a statement is true means to prove the inclusion A 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. The well-known Weierstrass example of a continuous but nowhere differentiable function f is a counterexample to include A B, since f is an element of A that does not belong to B. At the risk of oversimplification, we can say that mathematics (except for definitions, statements and calculations) consists of two parts - proofs and counterexamples, and mathematical discoveries consist in finding proofs and constructing counterexamples.
This determines the relevance of counterexamples during the formation and development of mathematics.
Most mathematical books are devoted to proving true statements.
Generally speaking, there are two types of examples in mathematics - illustrative examples and counterexamples. The first show why this or that statement makes sense, and the second - why this or that statement is meaningless. It can be argued that any example is at the same time a counterexample to some assertion, namely, to the assertion that such an example is impossible. We do not wish to give the term counterexample such a universal meaning, but we assume 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. Similarly, the class of all monotone functions on a bounded closed interval as a class of integrable functions is not a counterexample, but the same class, as an example of a function space, but not a vector space, is a counterexample.
The purpose of this paper 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 analysis
2. Define the notion 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 the function
6. Consider the necessary condition for a local extremum
7. Consider sufficient conditions for a local extremum
1. Counterexamples in analysis
1.1. The notion of a counterexample
Popular expressions: "learn from examples", "the power of example" have not only worldly meaning. The word "example" is cognate with the words "measure", "measure", "measure", but not only for this reason is present in mathematics from its very beginnings. The 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 value.
A counterexample is an example that refutes the truth of some statement.
Constructing a counterexample is a common way of refuting hypotheses. If there is a statement like "For any X from the set M, property A holds," then the counterexample for this statement is any object X 0 from the set M for which property A does not hold.
A classic counterexample in the history of calculus is the function constructed by Bernard Bolzano, which is continuous on the entire real axis and is not differentiable at any point. This function served as a counterexample to the hypothesis that the 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 apply to infinite limits as well.
However, the term differentiable function is used only if the function has a finite derivative at every point of its domain. 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 or exp(x).
It is assumed that all sets, including domains and sets of values of functions, are subsets of R. Otherwise, a corresponding refinement will be made.
1. A 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 to take all intermediate values, and this property is inherent not only in continuous functions, but also in derivatives ( see, p. 84, exercise 40, and also, vol. I, p. 224). The following is an example of a discontinuous derivative.
2. Differentiable function with discontinuous derivative
Consider the function
Its derivative
is discontinuous at the point x = 0.
3. A discontinuous function that has a derivative everywhere (not necessarily finite)
In order to make such an example possible, the definition of the derivative must be extended to include the values ± . Then the discontinuous function sgn x (Example 1) has a derivative
4. A differentiable function whose derivative does not preserve sign in any one-sided neighborhood of an 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 monotone in any one-sided neighborhood of the point x = 0.
5. A differentiable function whose derivative is positive on 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 bounded on a closed interval
Consider the function
Its derivative
not limited to [-1, 1].
7. A function whose derivative exists and is bounded, but has no (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 | everywhere continuous, but not differentiable at the point x - 0. Using the 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 subsection, we will give an example using an infinite set of shifts of the function | x |. Let us show that the function nowhere differentiable. Let a be an arbitrary real number, and let for any natural n the number h n equal to 4 -n or –4 -n be chosen so that Then the value has the same value | h n | for all m n and is zero for m > n. Then the difference ratio is an integer that is even for even n and odd for odd n. It follows from this that the limit does not exist, and therefore does not exist and The example given is a modification of the example built by B. L. Van der Waerden in 1930 (see, p. 394). The very first example of a continuous nowhere differentiable function was constructed by K. W. T. Weierstrass (German mathematician, 1815-1897): where a is an odd integer and b is such that At present, 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 also in (vol. II, pp. 401-412). The function of this example is not monotonic on any interval. Moreover, there is an example of a function that is everywhere differentiable and nowhere monotone (see Vol. II, pp. 412-421). The construction of this example is very complicated 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 turn to a complex-valued function. Function real variable x is everywhere continuous and differentiable (see, pp. 509-513). However, there is no such interval for which, for some, 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 p. 78), we have a contradiction. 13. An infinitely differentiable monotonic 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 monotonic function with the indicated property. We set f(x) 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 multiplying by appropriate negative factors. 2.1. Monotonicity of functions The function f (x) is called 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). The function f (x) is called 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). Picture 1. In the graph shown in the figure, the function y \u003d f (x), increases on each of the intervals [ a ; x 1) and (x 2 ; b ] and decreasing on the interval (x 1 ; x 2). Note that the function is increasing on each of the intervals [ a ; x 1) and (x 2 ; b ], but not on the union gaps If a function is increasing or decreasing on some interval, then it is called monotonic on this interval. Note that if f is a monotonic 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. We list the properties of monotone functions (we assume that all functions are defined on some interval D). Similar assertions can also be made for a decreasing function. Rice. 2. Function properties. A point a is called a point of maximum of a function f if there is such an ε-neighborhood of the point a that for any x from this neighborhood the inequality f (a) ≥ f (x) holds. A point a is called a minimum point of the function f if there is such an ε-neighborhood of the point a that for any x from this neighborhood the inequality f (a) ≤ f (x) is satisfied. 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. So, 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 an internal point of the domain of definition. If for any (x ≠ a) the inequality f (x) ≤ f (a) is satisfied, then the point a is called the point of the greatest value of the function on the set D: If for any (x ≠ b) the inequality f (x) > f (b) is satisfied, then the point b is called the point of least value of the function on the set D .2. Monotone functions
