Let's construct an auxiliary function on a segment step by step. At the zero step we will set two points:

And .

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 along 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:

And , etc.

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 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 monotonic, 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.

MINISTRY OF EDUCATION AND SCIENCE OF THE RUSSIAN FEDERATION

PUBLIC EDUCATIONAL INSTITUTION

HIGHER PROFESSIONAL EDUCATION

"USSURI STATE PEDAGOGICAL INSTITUTE"

Faculty of Physics and Mathematics

Coursework on mathematical analysis

Topic: “Continuous but not differentiable functions”

Completed by: Plyasheshnik Ksenia

student of group 131

Head: Delyukova Y.V.

Ussuriysk - 2011

Introduction........................................................ ........................................... 3

Historical background................................................... ......................... 4

Basic definitions and theorems................................................................. ....... 5

Example continuous function without derivative........................... 10

Solution to the exercises................................................... ........................ 13

Conclusion................................................. ..................................... 21

References........................................................ ........................... 22

Introduction

The course work is devoted to the study of the connection 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, constructed by van der Waerden;

3. Decide on the exercise system.

Historical background

Bartel Leendert van der Waerden (Dutch Bartel Leendert van der Waerden, February 2, 1903, Amsterdam, Netherlands - January 12, 1996, Zurich, Switzerland) - Dutch mathematician.

He studied at the University of Amsterdam, then at the University of Gottingen, where Emmy Noether had a huge influence on him.

Major works in the field of algebra, algebraic geometry, where he (along with Andre Weil and O. Zariski) raised the level of rigor, and mathematical physics, where he worked on the application of group theory to questions quantum mechanics(along with Hermann Weyl and Yu. Wigner). His classic book Modern Algebra (1930) became the model for subsequent textbooks on abstract algebra and went through many editions.

Van der Waerden is one of the leading specialists in the history of mathematics and astronomy in Ancient world. His Awakening Science (Ontwakende wetenschap 1950, Russian translation 1959) gives an extensive 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 presentation of Pythagorean views on musical harmony.

Basic definitions and theorems

Limit of a function at a point. Left and right limits

Definition (Cauchy limit, in the language Number is called the limit of a function at a point if

Definition (in neighborhood language) A number is called the limit of a function at a point if for any -neighborhood of the number there exists a -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 (that is, the corresponding sequence of function values ​​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 existence of a limit)

In order for a limit of a function to exist at a point, it is necessary and sufficient that there be left and right limits that are equal to each other.

The concept of derivative. One-sided derivatives.

Consider a function defined on the set

1. Let's take the increment. Let's give the point an increment. We get .

2. Let's calculate the value of the function at points. And

3. .

4. .

and the increment of the argument can be either positive or negative, then this limit is called the derivative at a point and is denoted by . It can also be infinite.

left (left-sided) 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 the point:

( ( .

Consider the function Let's find one-sided derivatives at the point

Hence, ( =-1; ( =1 And ( ( , that is, the function has no derivative at a point.

Various definitions continuity of a function at a point.

Definition 1 (main) A function is called continuous at the point if the limit of the function at equal to the value functions at this point.

Definition 2 (in language A function is called continuous at a point if ε, δ>0, such that .

Definition 3 (according to Heine, in sequence language) 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 the 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 is const, independent of , is infinitesimal at

Definition 2 A function that is differentiable at any point in the 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

Converse theorem. If a function is continuous, then it is differentiable.

The converse theorem is not true.

B is not differentiable, although continuous.

Classification of break points

Definition A function that is not continuous at a point is discontinuous at the point, and the point itself is called a discontinuity point.

There are two classifications of break points: type I and type II.

Definition A point is called a discontinuity point of the first kind if at this point there are finite one-sided limits unequal to each other.

Definition A point is called a point of removable gap yva, if , but they are not equal to the value of the function at the 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;

· – infinite or – endless;

Signs uniform convergence next to V

Weierstrass sign.

If members functional range(1) satisfy in the domain the inequalities where is a term of some convergent number series then series (1) converges uniformly.

Theorem 1 Let the functions are defined in an interval and are all continuous at some point in this interval. If series (1) converges uniformly in the interval, then the sum of the series at the point will also be continuous.

Example of a continuous function without derivative

The first example of this kind was built by Weierstrass; its function is defined as follows:

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 was able to show that, nevertheless, there is no finite derivative for it at any point.

Here we will consider a simpler example by van der Waerden, built essentially on the same idea, only the oscillating curves y = cosωχ are replaced by oscillating broken lines.

So, let's denote by absolute value the difference between the number χ and the nearest integer. 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 broken line have an angular coefficient of ±1.

Let us then assume for k=1,2,3,…:

This function will be linear in intervals of the form ; it is also continuous and has a period. Its graph is also broken, but with smaller teeth; Fig. 1(b), for example, shows a graph of the function . In all cases slope coefficients individual links of the broken line and here are equal to ±1.

Let us now define, for all real values ​​of x, the function f (x) by the equality

Since, obviously, 0≤ (k =0,1,2,...), so that the series is majorized by the convergent progression, then (as in the case of the Weierstrass function) the series converges uniformly, and the function is continuous everywhere.

Let's stop at any value. Calculating it to within (where n =0,1,2,...), by deficiency and excess, we will enclose 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 within the other. In each of them there is a point such that its distance from the point is equal to half the length of the interval.

It is clear that as n increases, options .

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 a function that is linear in the interval will also 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 here it is clear that when the ratio of increments to any finite limit tends cannot, so our function has no finite derivative.

Solution of exercises

Exercise 1 (, No. 909)

The function is defined as follows: . Explore 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 rupture

Since the left limit is equal to the right limit and 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, let us assume the opposite. 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.

Method 2. 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 discontinuity at the point

Because - infinitely small function, - limited.

Let us prove that the function has no limit at the point.

To prove it, it is enough 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 (, No. 995)

Show that the function where is a continuous function and has no derivative at the point . What are one-sided derivatives equal to?

One-sided limits are not equal; the function does not have a derivative at the point.

Exercise 4 (, No. 996)

Construct an example of a continuous function that does not have a derivative function at given points:

Consider the 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 (, No. 125)

Show that the function has no derivative at the point.

Let's find the increment of the function at the point

Let's create the ratio of the increment of a function at a point to the increment of the argument

Let's go to the limit

Exercise 6 (, №128)

Show that function has no derivative at the point.

Let's take the increment Let's give the point an increment We'll get

Let's find the value of the function at points and

Let's find the increment of the function at the point

Let's create the ratio of the increment of a function at a point to the increment of the argument

Let's go to the limit

Conclusion: does not have a finite derivative at the point.

Exercise 7 (, №131)

Examine a function for continuity

– point suspicious for rupture

Since the left limit is equal to the value of the function at a point, then the function is continuous at the point and there is a discontinuity of the first kind.

Conclusion

IN course work material related to the concept of “Continuous but not differentiable functions” is presented, the goals of this work have been achieved, the problems have been solved.

References

1. B. P. Demidovich, / Collection of problems for the course of mathematical analysis. Tutorial for students of the Faculty of Physics and Mathematics pedagogical institutes. – M.: Education, 1990 –624 p.

2. G. N. Berman, / Collection of problems for the course of mathematical analysis. – M.: Nauka, 1977 – 416 p.

3. G. M. Fikhtengolts, / Course of differential and integral calculus Vol. II. - M., Science, 1970-800s.

4. I.A. Vinogradova, /Tasks and exercises in mathematical analysis, part 1. – M.: Bustard, 2001 – 725 p.

5. Internet resource \ http://ru.wikipedia.org/wiki.

6. Internet resource \http://www.mathelp.spb.ru/ma.htm.

Let's construct an interesting set on the plane IN as follows: divide, square by straight lines
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:

The Sierpinski Cemetery is a perfect and nowhere dense multitude.

Let us note the fractal structure of the set.

2.2 Cantor's comb

Let's call Cantor comb many D on the plane Oxy, consisting of all points
, 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.

Is it possible to set B(Sierpinski Cemetery) and D(Cantor comb) express through the Cantor set
using the operations of the segment's complement and the Cartesian product? It is obvious that the sets B And D expressed simply:

B=
x

D= x

3 Cantor function

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 obtained 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 \
. Constructed function
called Cantor function(Cantor function), and its graph below is ""devil's stairs"".

Note the fractal structure of the function:

Function
satisfies the following inequality:

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
has no jumps. And because a monotonic function cannot have discontinuity points other than jumps (see the continuity criterion monotonic functions), then it is continuous.

An interesting observation is that the graph of the continuous Cantor function
It is impossible to draw "without lifting the pencil from the paper."

1. A function that is continuous everywhere but differentiable nowhere

Let's build an auxiliary function
on a segment step by step. At the zero step we will set two points:

And
.

Next we fix the parameter . 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.

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 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 or above, 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 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
, 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 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 . By changing the value of this parameter, you can obtain families of functions with their own special properties.