The theory of functions of one change. Mathematical analysis. Theory of functions of one change Matan theory 1 semester
A.V. Glasko
LECTURE from mathematical analysis
"ELEMENTARY FUNCTIONS I INTERIOR"
Moscow, MSTU im. Not. Bauman
§1. Logic symbolism.
When writing mathematical virases, we will use the following logical symbols:
value |
value |
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For be-like, for be-like, for all (see |
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Snu, know, є (exist) |
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Tyagne, slid (out) |
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Equivalently, todi and tilki todi, |
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necessary and sufficient |
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So if A and B be-yaky loving, then |
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value |
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A chi B (abo A abo B, abo і A і B) |
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For be-like x maє mice A |
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Isnuє x, for any ma mіsce A |
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Z A slid B (if it's true A, then it's true B) |
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(Іmplіkatsіya) |
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A is equivalent to B, A maє mice todi and only todі, if maє mіsce B, |
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for In necessary and sufficient A |
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Respect. "A B" means that for B to complete A, and for A it is necessary to B. |
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Butt. (X = 1) => (x2 -3x + 2 = 0) => ((x = 1) (x = 2)).
In some cases, we will use one special symbol: A = df B.
Vin means, scho A = B for the value.
§2. Without lichi. Elements and parts of the lifeless.
Understanding a lot is the first thing to understand, as it does not come from simple. Words: sukupnist, family, typ - yogo synonyms.
Add a lot: without students in the classroom, without children in the department, without cars in the parking lot and in.
The first to understand it is also є the understanding elementlessі vіdnosini
mіzh elements without lіchі.
Butt. N - no natural numbers, where elements є are numbers 1,2,3, ... If x and y are elements of N, then the stench is found in one of the next steps: x = y, x
We are familiar with the meaningless by the great letters: A, B, C, X, Y, ..., and the elements are malimi: a, b, c, x, y, ...
Vіdnosini mіzh elements or plurals are represented by symbols inserted by letters. Forward. Nekhai A is a deyakiy bezl_ch. Todi vidnoshennya a A means, who a is an element without A.
You can ask for nothing in different ways... 1. Pererahuvannyam yogo elements.
For example, A = (a, b, c, d), B = (1, 7, 10)
2. Vkaz_vkoyu power of elements. Nekhai A - bezlich elements a, who will power the r. The price can be written at the viewer: A = (a: p) or A = (ap).
For example, writing A = (x: (x R) (x2 -1> 0)) means that A is without any valid numbers, which is happy with the irregularities x2 -1> 0.
Enter a number of important values.
Def. Bezlich is called Kintsev, as it is a singular number of elements. In the first place, it is not called endless.
For example, the lack of students in the classroom is uncharacteristic, and the lack of natural numbers or the number of points in the middle is endless.
Def. Bezlich, how not to take revenge on the desired element, to be called empty and to be known.
Def. Two desperate ones are called pryvnyi, as the stench is stored from one and the same
Too, the understanding of the helpless is not able to respect the order of the elements. Def. Bezlich X is called a sub-multiplicity of Y, as if it were an element of a powerless X є an element of a multiplicity Y
element of Y є element of X). When tsyom vikoristovutsya meaning: X Y.
For example, without oranges O є by a multiplicity of fruits F: O F, and without natural numbers N є by a multiplicity of numbers R: N R.
The symbol "" i "" is called inclusion symbols. Vvazhayut, scho skinless є to myself. Empty a lot є be a lot.
Def. Be-yak not empty pidmzhina In without A, not equal to A, be called
Let's get out of the way.
§ 3. Diagrams of Euler-Venn. Elementary operations on multiples.
It is impossible to visualize it graphically by hand, at the sight of areas on the square. At the same time, we rely on respect, so that the points of the region correspond to the elements of many. Such graphical statements are called Euler-Venn diagrams.
Butt. A - without students at the Moscow State Technical University, B - without students in the classroom. Small. 1 demonstrating on purpose A B.
Diagrams of Euler-Venn manually vikoristovuvati for the scientific image of the elementary operation on multiples... Prior to the main operations, these are the following.
Small. 1. The butt of the Euler-Venn diagrams.
1. Overreturns A B set A and B is called a set C, which can be stored from all the elements, but one hour overlaps with sets A and B:
C = A B = df (z: (z A) (z B))
(In Fig. 2, no C is represented by the shaded area).
Small. 2. Peretin mnogozhin.
2.About A B set A and B are called set C, which is stored from all elements, so there is a wish to have one set A and B.
C = A B = df (z: (z A) (z B))
(In Fig. 3, no C is represented by the shaded area).
Small. 3. Ob'dnannya plural.
Small. 4. Riznitsya plural.
3. The difference A \ B of the set A and B is called the set C, which can be stacked with few elements, which can be without A, or not be without B:
A \ B = (z: (z A) (z B))
(In Fig. 4, the bezl_ch C is represented by the region filled with zhovty color).
§4. The set of valid numbers.
Let me wake up without words of speech (action) numbers R. without natural numbers, Yake is a significant offensive rank. In the quality of the first element, the number n = 1 is necessary. The skin offensive element will be trimmed out of the following units:
N = (1, 1 + 1, (1 + 1) +1, ...) = (1, 2, 3, ..., n, ...).
N = (-1, -2, -3, ..., -n, ...).
Without whole numbers Z It is meaningful that there are three multiples: N, -N і without lіchі, which can be stored from one element - zero:
Without any racial numbers, it is significant as without any number of different numbers:
Q = (xx = m / n; m, n Z, n 0).
Obviously, N Z Q.
Seemingly, the skin-rational number can be recorded in the view of the endless action or the non-terminated periodic fraction. How many rational numbers are there for the definition of all the values that can be created when we are awakened? Already in Ancient Greece, it is shown that it’s dumb rectangular tricycle With the legs, one can not imagine a racial number in the view of the hypotenuse. With such a rank, we cannot be contained without rational numbers. It is necessary to expand the understanding of numbers. Expansion to reach inputs without random numbers J, as simple as mislity as without all non-recurring non-endless dozens of fractions.
Adding a set of rational and rational numbers are called
without any action (speech) numbers R: R = Q Y.
In some cases, look at a wider range of unlimited R numbers, more reasonable
Show the numbers manually with dots on the number axis.
Def. The numerical view is called straight, on which the ear is indicated, the scale is directly shown.
Between the numbers and the points of the numerical axis, the answer is unambiguous: whether any material number is displayed, a single point of the numerical axis and navpaki.
Axiom of non-interruptions (non-interruptions) of arbitrary numbers. If you want to be biased boules are not empty without A = (a) R і B = (b) R so, for any a and b you will see the inability a ≤ b, you know there is a number cR is also where a ≤ c ≤ b (Fig. 5).
Fig. 5. Illustration of the axiom of the use of meaningless numbers.
§5. Multiple numbers. About.
Def. numerically without be called be-yak pіdnіlіchі R. Nayvazhіvіnіnі nіlіchі: N, Z, Q, J, as well as
download: (x R | a x b),
interval: (a, b) (x R | a x b), (,) = R
interval: (x R | a x b),
(X R | x b).
I will play a role in the mathematical analysis of the eye of the eye of the point of the numerical axis.
Def. -neighborhood of point x 0 is called an interval up to 2 with center at point x 0 (Fig. 6):
u (x 0) (x 0, x 0).
Small. 6. Near the point.
Def. The punctured neighborhood of the point is called the neighborhood of the center of the point,
the point x 0 itself is enabled (fig. 7):
u (x 0) u (x 0) \ (x 0) (x 0, x 0) (x 0, x 0).
Small. 7.Puncture near the point.
Def. Right-handed -neighborhood of point x0 be called napivinterval
u (x 0), range of value: E = [-π / 2, π / 2].
Small. 11. Graph of functions y arcsin x.
Introduced now folding function (composition of the image). Let there be given three free images D, E, M and let f: D → E, g: E → M. Obviously, it is possible to create a new image h: D → M, called the composite image f and g, or a folding function (Fig. 12).
The folding function is called the offensive rank: z = h (x) = g (f (x)) or h = f o g.
Small. 12. Illustration to the understanding of the folding function.
The function f (x) is called internal function, And the function g (y) - new functions.
1. Internal function f (x) = x², value g (y) sin y. Folding function z = g (f (x)) = sin (x²)
2. Now navpaki. Internal function f (x) = sinx, value g (y) y 2.u = f (g (x)) = sin² (x)
Let the size change x n accept an indefinite value
x 1 , x 2 , ..., x n , ..., (1)
and why the law of change of change x n, Tobto for skin natural number n can be specified x n... In such a rank, it is possible to transfer x nє function from n:
x n = F (n)
Of course, one of the most important to understand mathematical analysis is the line between the last values, or, well, the same, between the different magnitudes x n, Try the last x 1 , x 2 , ..., x n , ... . .
Viznachennya. any number a be called borderline x 1 , x 2 , ..., x n , ... . for the boundary wintry x n, For yak, for an appropriately small positive number e, there is also a natural number N(Tobto number N), x n, fixing it x N, seen from a on absolute value less, lower by e. given value briefly write like this:
| x n - a |< (2)
at all n N, Abo, well the same,
Viznachennya between according to Koshi... The number A is called the boundary of the function f (x) at point a, since the function is designated in the case of the point a behind a vignette, perhaps the point a itself, і for skin ε> 0 іnuє δ> 0 also, for all x, well please mind | x - a |< δ, x ≠ a, выполняется неравенство |f (x) – A| < ε.
Viznachennya between Heine... The number A is called the boundary of the function f (x) at point a, since the function is designated in the action near point a behind the vignette, perhaps, the point a itself, and for whether it is 
converging to the number a, it is likely that the value of the function converges to the number A.
As the function f (x) is located at point a, then the line is between.
The number A 1 is called the boundary function f (x) evil at point a, if for skin ε> 0 ісує δ> 
The number A 2 is called the boundary function f (x) on the right at the point a, if for skin ε> 0 іsnuє δ> 0 also, for all of them there is an inaccuracy 
Between evil is meant between right-handed person - The integer characterizes the behavior of the function of evil and right-handed person from point a. Їх are often called one-sided lines. In the designated one-sided between at x → 0, you want to drop the first zero: i. So, for the function 
Also for skin ε> 0 ісу such δ-okіl point a, but for all x, which are satisfied with | x - a |< δ, x ≠ a, выполняется неравенство |f (x)| >ε, then it seems that the function f (x) has an unlimited boundary at point a:
So, the function is at the point x = 0 with an unlimited boundary. So,
Even for skin ε> 0 іsny also δ> 0, but for any x> δ to see the irregularity | f (x) - A |< ε, то говорят, что предел функции f (x) при x, стремящемся к плюс бесконечности, равен A:
Theorem about the exact upper bound
value:АR mR, m - upper (lower) face of А, when аА аm (аm).
value: Bezlich A is surrounded by above (below), like іsnu takе m, scho аА, vikonutsya аm (аm).
value: SupA = m, where 1) m is the supremum of A
2) m ': m'
InfA = n, where 1) n is the lower edge of A
2) n ': n'> n => n "is not the lower boundary of A
value: SupA = m is a number, as follows: 1) aA am
2) > 0 a A, also a a-
InfA = n is a number, as follows: 1) 1) aA an
2) > 0 a A, also, a E a +
theorem: Be-yake, neporozhnє surrounded from the top without lich АR, has the exact upper edge, with only one.
Delivered:
Let's move on a numerical straight line the number m and the upper bound A.
[M] = max ([a]: aA) [[m], [m] +1] A => [m] +1 is the upper edge of A
Відрізок [[m], [m] +1] - split into 10 parts
m 1 = max: aA)]
m 2 = max, m 1: aA)]
m k = max, m 1 ... m K-1: aA)]
[[M], m 1 ... m K, [m], m 1 ... m K + 1/10 K] A => [m], m 1 ... m K + 1 / 10 K - top edge A
Arguably, m = [m], m 1 ... m K is the exact upper bound and і wona din:
k :)