Theory of functions of one variable. Mathematical analysis

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: AR mR, m - upper (lower) face of A, if аА аm (аm).

Definition: The set A is bounded from above (from below), if there exists m such that аА, then аm (аm) is satisfied.

Definition: SupA=m, if 1) m - upper bound of A

2) m’: m’ m' is not an upper face of A

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)  aA am

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 АR 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]:aA) [[m],[m]+1]A=>[m]+1 - upper face of A

Segment [[m],[m]+1] - split into 10 parts

m 1 =max:aA)]

m 2 =max,m 1:aA)]

m to =max,m 1 ...m K-1:aA)]

[[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: .

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)