A.V. Glasco
LECTURES ON MATHEMATICAL ANALYSIS
"ELEMENTARY FUNCTIONS AND LIMITS"
Moscow, MSTU im. N.E. Bauman
§one. logical symbolism.
When writing mathematical expressions, we will use the following logical symbols:
Meaning |
Meaning |
||
For anyone, for everyone, for everyone (from |
|||
There is, there is, there is (exist) |
|||
entails, follows (therefore) |
|||
Equivalently, if and only if, |
|||
necessary and sufficient |
|||
So if A and B are any propositions, then |
|||
Meaning |
|||
A or B (or A or B, or both A and B) |
|||
For any x we have A |
|||
There is x for which A holds |
|||
From A follows B (if A is true, then B is true) |
|||
(implication) |
|||
A is equivalent to B, A occurs if and only if B occurs, |
|||
A is necessary and sufficient for B |
|||
Comment. “A B” means that A is sufficient for B and B is necessary for A. |
|||
Example. (x=1) => (x2 -3x+2=0) => ((x=1) (x=2)).
Sometimes we will use another special character: A =df B.
It means that A = B by definition.
§2. Sets. Elements and parts of a set.
The concept of a set is a primary concept, not defined in terms of simpler ones. The words: set, family, set are its synonyms.
Examples of sets: many students in the classroom, many teachers in the department, many cars in the parking lot, etc.
Primary concepts are also the concepts set element and relationships
between the elements of the set.
Example. N is the set of natural numbers, its elements are the numbers 1,2,3, ... If x and y are elements of N, then they are in one of the following relations: x = y, x
We agree to denote sets by capital letters: A, B, C, X, Y, …, and their elements by lowercase letters: a, b, c, x, y, …
Relationships between elements or sets are indicated by symbols inserted between letters. For example. Let A be some set. Then the relation a A means that a is an element of the set A. The notation a A means that a is not an element of A.
The set can be defined in various ways. 1. Enumeration of its elements.
For example, A=(a, b, c, d), B=(1, 7, 10)
2. Specifying the properties of the elements. Let A be the set of elements a with property p. This can be written as: A=( a:p ) or A=( ap ).
For example, the notation А= ( x: (x R ) (x2 -1>0) ) means that A is a set of real numbers satisfying the inequality x2 -1>0.
Let us introduce some important definitions.
Def. A set is called finite if it consists of a certain finite number of elements. Otherwise, it is called infinite.
For example, the set of students in the classroom is finite, but the set of natural numbers or the set of points inside the segment is infinite.
Def. A set that does not contain any element is called empty and is denoted.
Def. Two sets are said to be equal if they consist of the same
Those. the concept of a set does not imply a particular order of elements. Def. A set X is called a subset of a set Y if any element of the set X is an element of the set Y (in this case, generally speaking, not any
an element of the set Y is an element of the set X). In this case, the designation is used: X Y.
For example, the set of oranges O is a subset of the set of fruits F : O F , and the set of natural numbers N is a subset of the set of real numbers R : N R .
The characters “ ” and “ ” are called inclusion characters. Each set is considered to be a subset of itself. The empty set is a subset of any set.
Def. Any non-empty subset B of a set A that is not equal to A is called
own subset.
§ 3. Euler-Venn diagrams. Elementary operations on sets.
It is convenient to represent sets graphically, as regions on a plane. This implies that the points of the region correspond to the elements of the set. Such graphical representations of sets are called Euler-Venn diagrams.
Example. A is the set of MSTU students, B is the set of students in the audience. Rice. 1 clearly demonstrates that A B .
Euler-Venn diagrams are convenient to use for a visual representation of elementary operations on sets. The main operations include the following.
Rice. 1. An example of an Euler-Venn diagram.
1. The intersection A B of sets A and B is the set C, which consists of all elements belonging simultaneously to both sets A and B:
C=A B =df ( z: (z A) (z B) )
(in Fig. 2, the set C is represented by the shaded area).
Rice. 2. Intersection of sets.
2. The union A B of sets A and B is the set C, which consists of all elements belonging to at least one of the sets A or B.
C=A B =df ( z: (z A) (z B) )
(in Fig. 3, the set C is represented by the shaded area).
Rice. 3. Union of sets.
Rice. 4. Difference of sets.
3. The difference A \ B of sets A and B is the set C, consisting of all elements belonging to set A, but not belonging to set B:
A \ B =( z: (z A) (z B) )
(in Fig. 4, the set C is represented by the area shaded in yellow).
§4. The set of real numbers.
Let us construct a set of real (real) numbers R. To do this, consider, first of all, set of natural numbers, which we define as follows. Let's take the number n=1 as the first element. Each subsequent element will be obtained from the previous one by adding one:
N = (1, 1+1, (1+1)+1, …) = ( 1, 2, 3, …, n, … ).
N = ( -1, -2, -3, ..., -n, ... ).
The set of integers Z define as the union of three sets: N, -N and a set consisting of a single element - zero:
The set of rational numbers is defined as the set of all possible ratios of integers:
Q = ( xx = m/n; m, n Z, n 0 ).
Obviously, N Z Q.
It is known that every rational number can be written as a finite real or infinite periodic fraction. Are rational numbers sufficient to measure all the quantities that we can meet in the study of the world around us? Already in Ancient Greece it was shown that it is not: if we consider an isosceles right triangle with legs of length one, the length of the hypotenuse cannot be represented as a rational number. Thus, we cannot restrict ourselves to the set of rational numbers. It is necessary to expand the concept of number. This extension is achieved by introducing sets of irrational numbers J, which is easiest to think of as the set of all non-periodic infinite decimals.
The union of sets of rational and irrational numbers is called
set of real (real) numbers R: R =Q Y.
Sometimes they consider an extended set of real numbers R, understanding
Real numbers are conveniently represented as dots on the number line.
Def. The numerical axis is called a straight line, which indicates the origin, scale and direction of reference.
A one-to-one correspondence is established between real numbers and points of the numerical axis: any real number corresponds to a single point of the numerical axis and vice versa.
Axiom of completeness (continuity) of the set of real numbers. Whatever non-empty sets А= ( a ) R and B= (b) R are such that for any a and b the inequality a ≤ b is true, there is a number cR such that a ≤ c ≤ b (Fig. 5).
Fig.5. Illustration of the axiom of the completeness of the set of real numbers.
§5. Numeric sets. Neighborhood.
Def. Numerical set any subset of the set R is called. The most important numerical sets: N, Z, Q, J, and also
segment: (x R | a x b ),
interval: (a ,b ) (x R |a x b ), (,)=R
half-intervals: ( x R| a x b),
(x R | x b ).
The most important role in mathematical analysis is played by the concept of a neighborhood of a point on the numerical axis.
Def. -neighborhood of the point x 0 is an interval of length 2 centered at the point x 0 (Fig. 6):
u (x 0 ) (x 0 ,x 0 ).
Rice. 6. Neighborhood of a point.
Def. The punctured -neighborhood of a point is the neighborhood of this point,
from which the point x 0 itself is excluded (Fig. 7):
u (x 0 ) u (x 0 )\(x 0 ) (x 0 ,x 0 ) (x 0 ,x 0 ).
Rice. 7. Punctured neighborhood of a point.
Def. The right-hand neighborhood of the point x0 called a half-interval
u (x 0 ) , range: E= [-π/2,π/2 ].
Rice. 11. Graph of the function y arcsin x.
Let us now introduce the concept of a complex function ( display compositions). Let three sets D, E, M be given and let f: D→E, g: E→M. Obviously, it is possible to construct a new mapping h: D→M, called a composition of mappings f and g or a complex function (Fig. 12).
A complex function is denoted as follows: z =h(x)=g(f(x)) or h = f o g.
Rice. 12. Illustration for the concept of a complex function.
The function f (x) is called internal function, and the function g ( y ) - external function.
1. Internal function f (x) = x², external g (y) sin y. Complex function z= g(f(x))=sin(x²)
2. Now vice versa. Inner function f (x)= sinx, outer g (y) y 2 . u=f(g(x))=sin²(x)
Let the variable x n takes an infinite sequence of values
x 1 , x 2 , ..., x n , ..., (1)
and the law of change of the variable is known x n, i.e. for every natural number n you can specify the corresponding value x n. Thus it is assumed that the variable x n is a function of n:
x n = f(n)
Let us define one of the most important concepts of mathematical analysis - the limit of a sequence, or, what is the same, the limit of a variable x n running sequence x 1 , x 2 , ..., x n , ... . .
Definition. constant number a called sequence limit x 1 , x 2 , ..., x n , ... . or the limit of a variable x n, if for an arbitrarily small positive number e there exists such a natural number N(i.e. number N) that all values of the variable x n, beginning with x N, differ from a less in absolute value than e. This definition is briefly written as follows:
| x n -a |< (2)
for all n N, or, which is the same,
Definition of the Cauchy limit. A number A is called the limit of a function f (x) at a point a if this function is defined in some neighborhood of the point a, except perhaps for the point a itself, and for each ε > 0 there exists δ > 0 such that for all x satisfying condition |x – a|< δ, x ≠ a, выполняется неравенство |f (x) – A| < ε.
Definition of the Heine limit. A number A is called the limit of a function f (x) at a point a if this function is defined in some neighborhood of the point a, except perhaps for the point a itself, and for any sequence such that 
converging to the number a, the corresponding sequence of values of the function converges to the number A.
If the function f(x) has a limit at the point a, then this limit is unique.
The number A 1 is called the left limit of the function f (x) at the point a if for each ε > 0 there exists δ > 
The number A 2 is called the right limit of the function f (x) at the point a if for each ε > 0 there exists δ > 0 such that the inequality 
The limit on the left is denoted as the limit on the right - These limits characterize the behavior of the function to the left and right of the point a. They are often referred to as one-way limits. In the notation of one-sided limits as x → 0, the first zero is usually omitted: and . So, for the function 
If for each ε > 0 there exists a δ-neighborhood of a point a such that for all x satisfying the condition |x – a|< δ, x ≠ a, выполняется неравенство |f (x)| >ε, then we say that the function f (x) has an infinite limit at the point a:
Thus, the function has an infinite limit at the point x = 0. Limits equal to +∞ and –∞ are often distinguished. So,
If for each ε > 0 there exists δ > 0 such that for any x > δ the inequality |f (x) – A|< ε, то говорят, что предел функции f (x) при x, стремящемся к плюс бесконечности, равен A:
Existence theorem for the least upper bound
Definition: AR mR, m - upper (lower) face of A, if аА аm (аm).
Definition: The set A is bounded from above (from below), if there exists m such that аА, then аm (аm) is satisfied.
Definition: SupA=m, if 1) m - upper bound of A
2) m’: m’
InfA = n if 1) n is the infimum of A
2) n’: n’>n => n’ is not an infimum of A
Definition: SupA=m is a number such that: 1) aA am
2) >0 a A, such that a a-
InfA = n is called a number such that:
2) >0 a A, such that a E a+
Theorem: Any non-empty set АR bounded from above has a best upper bound, and a unique one at that.
Proof:
We construct a number m on the real line and prove that this is the least upper bound of A.
[m]=max([a]:aA) [[m],[m]+1]A=>[m]+1 - upper face of A
Segment [[m],[m]+1] - split into 10 parts
m 1 =max:aA)]
m 2 =max,m 1:aA)]
m to =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 face A
Let us prove that m=[m],m 1 ...m K is the least upper bound and that it is unique:
to: )
